{"id":31,"date":"2019-08-20T17:01:20","date_gmt":"2019-08-20T21:01:20","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/rational-expressions\/"},"modified":"2022-06-01T10:39:17","modified_gmt":"2022-06-01T14:39:17","slug":"rational-expressions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/rational-expressions\/","title":{"raw":"Rational Expressions","rendered":"Rational Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section students will:\n<ul>\n \t<li>Simplify rational expressions.<\/li>\n \t<li>Multiply rational expressions.<\/li>\n \t<li>Divide rational expressions.<\/li>\n \t<li>Add and subtract rational expressions.<\/li>\n \t<li>Simplify complex rational expressions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167339311757\">A pastry shop has fixed costs of[latex]\\,\\text{\\$}280\\,[\/latex]per week and variable costs of[latex]\\,\\text{\\$}9\\,[\/latex]per box of pastries. The shop\u2019s costs per week in terms of[latex]\\,x,[\/latex]the number of boxes made, is[latex]\\,280+9x.\\,[\/latex]We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.<\/p>\n\n<div id=\"fs-id1167339295641\" class=\"unnumbered aligncenter\">[latex]\\frac{280+9x}{x}[\/latex]<\/div>\n<p id=\"fs-id1167339261982\">Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.<\/p>\n\n<div id=\"fs-id1167339317968\" class=\"bc-section section\">\n<h3>Simplifying Rational Expressions<\/h3>\n<p id=\"fs-id1167339185999\">The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let\u2019s start with the rational expression shown.<\/p>\n\n<div id=\"fs-id1167339433455\" class=\"unnumbered aligncenter\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/div>\n<p id=\"fs-id1167339344788\">We can factor the numerator and denominator to rewrite the expression.<\/p>\n\n<div id=\"fs-id1167339429080\" class=\"unnumbered aligncenter\">[latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/div>\n<p id=\"fs-id1167339199768\">Then we can simplify that expression by canceling the common factor[latex]\\,\\left(x+4\\right).[\/latex]<\/p>\n\n<div id=\"fs-id1167339318196\" class=\"unnumbered aligncenter\">[latex]\\frac{x+4}{x+7}[\/latex]<\/div>\n<div id=\"fs-id1167339281673\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339240476\"><strong>Given a rational expression, simplify it.<\/strong><\/p>\n\n<ol id=\"fs-id1167339306312\" type=\"1\">\n \t<li>Factor the numerator and denominator.<\/li>\n \t<li>Cancel any common factors.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1167339303419\">\n<div id=\"fs-id1167339321615\">\n<h3>Simplifying Rational Expressions<\/h3>\n<p id=\"fs-id1167339197034\">Simplify[latex]\\,\\frac{{x}^{2}-9}{{x}^{2}+4x+3}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339232113\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339232113\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339232113\"]\n<div id=\"fs-id1167339155326\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lllll}\\frac{\\left(x+3\\right)\\left(x-3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x-3}{x+1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1167339331535\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339432914\">We can cancel the common factor because any expression divided by itself is equal to 1.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339324584\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339231124\"><strong>Can the[latex]\\,{x}^{2}\\,[\/latex]term be cancelled in <a class=\"autogenerated-content\" href=\"#Example_01_06_01\">(Figure)<\/a>?<\/strong><\/p>\n<p id=\"fs-id1167339281697\"><em>No. A factor is an expression that is multiplied by another expression. The[latex]\\,{x}^{2}\\,[\/latex]term is not a factor of the numerator or the denominator.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1167339154122\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_01\">\n<div id=\"fs-id1167339299924\">\n<p id=\"fs-id1167339299925\">Simplify[latex]\\,\\frac{x-6}{{x}^{2}-36}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339126332\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339126332\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339126332\"]\n<p id=\"fs-id1167339166294\">[latex]\\frac{1}{x+6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339165929\" class=\"bc-section section\">\n<h3>Multiplying Rational Expressions<\/h3>\n<p id=\"fs-id1167339149550\">Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.<\/p>\n\n<div id=\"fs-id1167339185386\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339156547\"><strong>Given two rational expressions, multiply them.<\/strong><\/p>\n\n<ol id=\"fs-id1167339344628\" type=\"1\">\n \t<li>Factor the numerator and denominator.<\/li>\n \t<li>Multiply the numerators.<\/li>\n \t<li>Multiply the denominators.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_02\" class=\"textbox examples\">\n<div id=\"fs-id1167339330081\">\n<div id=\"fs-id1167339296544\">\n<h3>Multiplying Rational Expressions<\/h3>\n<p id=\"fs-id1167339281353\">Multiply the rational expressions and show the product in simplest form:<\/p>\n\n<div id=\"fs-id1167339432908\" class=\"unnumbered aligncenter\">[latex]\\frac{\\left(x+5\\right)\\left(x-1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x-1\\right)}{\\left(x+5\\right)}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339228244\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339228244\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339228244\"]\n<div id=\"fs-id1167339184717\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lllll}\\frac{\\left(x+5\\right)\\left(x-1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x-1\\right)}{\\left(x+5\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor the numerator and denominator}.\\hfill \\\\ \\frac{\\left(x+5\\right)\\left(x-1\\right)\\left(2x-1\\right)}{3\\left(x+6\\right)\\left(x+5\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply numerators and denominators}.\\hfill \\\\ \\frac{\\overline{)\\left(x+5\\right)}\\left(x-1\\right)\\left(2x-1\\right)}{3\\left(x+6\\right)\\overline{)\\left(x+5\\right)}}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Cancel common factors to simplify}.\\hfill \\\\ \\frac{\\left(x-1\\right)\\left(2x-1\\right)}{3\\left(x+6\\right)} \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339281579\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_02\">\n<div id=\"fs-id1167339228502\">\n<p id=\"fs-id1167339228503\">Multiply the rational expressions and show the product in simplest form:<\/p>\n\n<div id=\"fs-id1167339228507\" class=\"unnumbered aligncenter\">[latex]\\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339228097\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339228097\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339228097\"]\n<p id=\"fs-id1167339212908\">[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339281343\" class=\"bc-section section\">\n<h3>Dividing Rational Expressions<\/h3>\n<p id=\"fs-id1167339185206\">Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite[latex]\\,\\frac{1}{x}\u00f7\\frac{{x}^{2}}{3}\\,[\/latex]as the product[latex]\\,\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}.\\,[\/latex]Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.<\/p>\n\n<div id=\"fs-id1167339344761\" class=\"unnumbered aligncenter\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\n<div id=\"fs-id1167339117616\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339344893\"><strong>Given two rational expressions, divide them.<\/strong><\/p>\n\n<ol id=\"fs-id1167339344897\" type=\"1\">\n \t<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\n \t<li>Factor the numerators and denominators.<\/li>\n \t<li>Multiply the numerators.<\/li>\n \t<li>Multiply the denominators.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_03\" class=\"textbox examples\">\n<div id=\"fs-id1167339296449\">\n<div id=\"fs-id1167339296451\">\n<h3>Dividing Rational Expressions<\/h3>\nDivide the rational expressions and express the quotient in simplest form:\n<div id=\"fs-id1167339281593\" class=\"unnumbered aligncenter\">[latex]\\frac{2{x}^{2}+x-6}{{x}^{2}-1}\u00f7\\frac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339185775\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339185775\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339185775\"]\n<div id=\"fs-id1167339185778\" class=\"unnumbered aligncenter\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}\u00f7\\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339137930\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_03\">\n<div id=\"fs-id1167339432773\">\n<p id=\"fs-id1167339432774\">Divide the rational expressions and express the quotient in simplest form:<\/p>\n\n<div id=\"fs-id1167339432777\" class=\"unnumbered aligncenter\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}\u00f7\\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339432756\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339432756\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339432756\"]\n<p id=\"fs-id1167339432757\">[latex]1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339429240\" class=\"bc-section section\">\n<h3>Adding and Subtracting Rational Expressions<\/h3>\n<p id=\"fs-id1167339429245\">Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let\u2019s look at an example of fraction addition.<\/p>\n\n<div id=\"fs-id1167339232046\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\frac{5}{24}+\\frac{1}{40}&amp; =&amp; \\frac{25}{120}+\\frac{3}{120}\\hfill \\\\ &amp; =&amp; \\frac{28}{120}\\hfill \\\\ &amp; =&amp; \\frac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339242302\">We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.<\/p>\n<p id=\"fs-id1167339242307\">The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were[latex]\\,\\left(x+3\\right)\\left(x+4\\right)\\,[\/latex]and[latex]\\,\\left(x+4\\right)\\left(x+5\\right),[\/latex]then the LCD would be[latex]\\,\\left(x+3\\right)\\left(x+4\\right)\\left(x+5\\right).[\/latex]<\/p>\n<p id=\"fs-id1167339212596\">Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of[latex]\\,\\left(x+3\\right)\\left(x+4\\right)\\,[\/latex]by[latex]\\,\\frac{x+5}{x+5}\\,[\/latex]and the expression with a denominator of[latex]\\,\\left(x+4\\right)\\left(x+5\\right)\\,[\/latex]by[latex]\\,\\frac{x+3}{x+3}.[\/latex]<\/p>\n\n<div id=\"fs-id1167339231542\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339231550\"><strong>Given two rational expressions, add or subtract them.<\/strong><\/p>\n\n<ol id=\"fs-id1167339231554\" type=\"1\">\n \t<li>Factor the numerator and denominator.<\/li>\n \t<li>Find the LCD of the expressions.<\/li>\n \t<li>Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\n \t<li>Add or subtract the numerators.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_04\" class=\"textbox examples\">\n<div id=\"fs-id1167339212435\">\n<div id=\"fs-id1167339228248\">\n<h3>Adding Rational Expressions<\/h3>\n<p id=\"fs-id1167339228254\">Add the rational expressions:<\/p>\n\n<div id=\"fs-id1167339228257\" class=\"unnumbered aligncenter\">[latex]\\frac{5}{x}+\\frac{6}{y}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339429820\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339429820\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339429820\"]\n<p id=\"fs-id1167339437713\">First, we have to find the LCD. In this case, the LCD will be[latex]\\,xy.\\,[\/latex]We then multiply each expression by the appropriate form of 1 to obtain[latex]\\,xy\\,[\/latex]as the denominator for each fraction.<\/p>\n\n<div id=\"fs-id1167339212482\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{5}{x}\\cdot \\frac{y}{y}+\\frac{6}{y}\\cdot \\frac{x}{x}\\\\ \\frac{5y}{xy}+\\frac{6x}{xy}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339331048\">Now that the expressions have the same denominator, we simply add the numerators to find the sum.<\/p>\n\n<div id=\"fs-id1167339331051\" class=\"unnumbered aligncenter\">[latex]\\frac{6x+5y}{xy}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1167339224193\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339212767\">Multiplying by[latex]\\,\\frac{y}{y}\\,[\/latex]or[latex]\\,\\frac{x}{x}\\,[\/latex]does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_06_05\" class=\"textbox examples\">\n<div id=\"fs-id1167339306374\">\n<div id=\"fs-id1167339306376\">\n<h3>Subtracting Rational Expressions<\/h3>\n<p id=\"fs-id1167339306381\">Subtract the rational expressions:<\/p>\n\n<div class=\"unnumbered\">[latex]\\frac{6}{{x}^{2}+4x+4}-\\frac{2}{{x}^{2}-4}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339435145\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339435145\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339435145\"]\n<div id=\"fs-id1167339435147\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}\\frac{6}{{\\left(x+2\\right)}^{2}}-\\frac{2}{\\left(x+2\\right)\\left(x-2\\right)}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Factor}.\\hfill \\\\ \\frac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\frac{x-2}{x-2}-\\frac{2}{\\left(x+2\\right)\\left(x-2\\right)}\\cdot \\frac{x+2}{x+2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply each fraction to get LCD as denominator}.\\hfill \\\\ \\frac{6\\left(x-2\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}-\\frac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply}.\\hfill \\\\ \\frac{6x-12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Apply distributive property}.\\hfill \\\\ \\frac{4x-16}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Subtract}.\\hfill \\\\ \\frac{4\\left(x-4\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243148\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339243155\"><strong>Do we have to use the LCD to add or subtract rational expressions?<\/strong><\/p>\n<p id=\"fs-id1167339243160\"><em>No. Any common denominator will work, but it is easiest to use the LCD.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1167339243167\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_04\">\n<div id=\"fs-id1167339243178\">\n<p id=\"fs-id1167339243180\">Subtract the rational expressions:[latex]\\,\\frac{3}{x+5}-\\frac{1}{x-3}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339432621\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339432621\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339432621\"]\n<p id=\"fs-id1167339432622\">[latex]\\frac{2\\left(x-7\\right)}{\\left(x+5\\right)\\left(x-3\\right)}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Simplifying Complex Rational Expressions<\/h3>\nA complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression[latex]\\,\\frac{a}{\\frac{1}{b}+c}\\,[\/latex]can be simplified by rewriting the numerator as the fraction[latex]\\,\\frac{a}{1}\\,[\/latex]and combining the expressions in the denominator as[latex]\\,\\frac{1+bc}{b}.\\,[\/latex]We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get[latex]\\,\\frac{a}{1}\\cdot \\frac{b}{1+bc},[\/latex]which is equal to[latex]\\,\\frac{ab}{1+bc}.[\/latex]\n<div id=\"fs-id1167339196667\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339196674\"><strong>Given a complex rational expression, simplify it.<\/strong><\/p>\n\n<ol id=\"fs-id1167339196678\" type=\"1\">\n \t<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n \t<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n \t<li>Rewrite as the numerator divided by the denominator.<\/li>\n \t<li>Rewrite as multiplication.<\/li>\n \t<li>Multiply.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1167339196712\">\n<div id=\"fs-id1167339196714\">\n<h3>Simplifying Complex Rational Expressions<\/h3>\n<p id=\"fs-id1167339196719\">Simplify:[latex]\\frac{y+\\frac{1}{x}}{\\frac{x}{y}}[\/latex].<\/p>\n\n<\/div>\n<div id=\"fs-id1167339259637\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339259637\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339259637\"]\n<p id=\"fs-id1167339259639\">Begin by combining the expressions in the numerator into one expression.<\/p>\n\n<div id=\"fs-id1167339259642\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}y\\cdot \\frac{x}{x}+\\frac{1}{x}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Multiply by }\\frac{x}{x}\\text{ to get LCD as denominator}.\\hfill \\\\ \\frac{xy}{x}+\\frac{1}{x}\\hfill &amp; \\\\ \\frac{xy+1}{x}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339306524\">Now the numerator is a single rational expression and the denominator is a single rational expression.<\/p>\n\n<div id=\"fs-id1167339306528\" class=\"unnumbered aligncenter\">[latex]\\frac{\\frac{xy+1}{x}}{\\frac{x}{y}}[\/latex]<\/div>\n<p id=\"fs-id1167339306577\">We can rewrite this as division, and then multiplication.<\/p>\n\n<div id=\"fs-id1167339306580\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}\\frac{xy+1}{x}\u00f7\\frac{x}{y}\\hfill &amp; \\\\ \\frac{xy+1}{x}\\cdot \\frac{y}{x}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\frac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223469\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_05\">\n<div id=\"fs-id1167339223480\">\n<p id=\"fs-id1167339223481\">Simplify:[latex]\\frac{\\frac{x}{y}-\\frac{y}{x}}{y}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339286463\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339286463\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339286463\"]\n<p id=\"fs-id1167339286464\">[latex]\\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339286517\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339286524\"><strong>Can a complex rational expression always be simplified?<\/strong><\/p>\n<p id=\"fs-id1167339286529\"><em>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1167339286537\" class=\"precalculus media\">\n<p id=\"fs-id1167339286544\">Access these online resources for additional instruction and practice with rational expressions.<\/p>\n\n<ul id=\"fs-id1167339286548\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/simpratexpress\">Simplify Rational Expressions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/multdivratex\">Multiply and Divide Rational Expressions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/addsubratex\">Add and Subtract Rational Expressions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/complexfract\">Simplify a Complex Fraction<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230898\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1167339230905\">\n \t<li>Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See <a class=\"autogenerated-content\" href=\"#Example_01_06_01\">(Figure)<\/a>.<\/li>\n \t<li>We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See <a class=\"autogenerated-content\" href=\"#Example_01_06_02\">(Figure)<\/a>.<\/li>\n \t<li>To divide rational expressions, multiply by the reciprocal of the second expression. See <a class=\"autogenerated-content\" href=\"#Example_01_06_03\">(Figure)<\/a>.<\/li>\n \t<li>Adding or subtracting rational expressions requires finding a common denominator. See <a class=\"autogenerated-content\" href=\"#Example_01_06_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_01_06_05\">(Figure)<\/a>.<\/li>\n \t<li>Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See <a class=\"autogenerated-content\" href=\"#Example_01_06_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167339230956\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1167339230960\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1167339230966\">\n<div id=\"fs-id1167339230967\">\n<p id=\"fs-id1167339230968\">How can you use factoring to simplify rational expressions?<\/p>\n\n<\/div>\n<div id=\"fs-id1167339230971\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339230971\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339230971\"]\n<p id=\"fs-id1167339230972\">You can factor the numerator and denominator to see if any of the terms can cancel one another out.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339230978\">How do you use the LCD to combine two rational expressions?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230982\">\n<div id=\"fs-id1167339230983\">\n<p id=\"fs-id1167339230984\">Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.<\/p>\n\n<\/div>\n<div id=\"fs-id1167339230988\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339230988\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339230988\"]\n<p id=\"fs-id1167339230989\">True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230995\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1167339231001\">For the following exercises, simplify the rational expressions.<\/p>\n\n<div id=\"fs-id1167339231004\">\n<div id=\"fs-id1167339231005\">\n<p id=\"fs-id1167339231006\">[latex]\\frac{{x}^{2}-16}{{x}^{2}-5x+4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339199459\">\n<div id=\"fs-id1167339199460\">\n<p id=\"fs-id1167339199461\">[latex]\\frac{{y}^{2}+10y+25}{{y}^{2}+11y+30}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339199522\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339199522\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339199522\"]\n<p id=\"fs-id1167339199523\">[latex]\\frac{y+5}{y+6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339199556\">\n<div id=\"fs-id1167339199557\">\n<p id=\"fs-id1167339199558\">[latex]\\frac{6{a}^{2}-24a+24}{6{a}^{2}-24}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220266\">\n<div id=\"fs-id1167339220267\">\n<p id=\"fs-id1167339220268\">[latex]\\frac{9{b}^{2}+18b+9}{3b+3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339220319\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339220319\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339220319\"]\n<p id=\"fs-id1167339220320\">[latex]3b+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220338\">\n<div id=\"fs-id1167339220339\">\n<p id=\"fs-id1167339220340\">[latex]\\frac{m-12}{{m}^{2}-144}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339428487\">\n<div id=\"fs-id1167339428488\">\n<p id=\"fs-id1167339428489\">[latex]\\frac{2{x}^{2}+7x-4}{4{x}^{2}+2x-2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339428554\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339428554\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339428554\"]\n<p id=\"fs-id1167339428556\">[latex]\\frac{x+4}{2x+2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339428591\">\n<div id=\"fs-id1167339428592\">\n<p id=\"fs-id1167339428593\">[latex]\\frac{6{x}^{2}+5x-4}{3{x}^{2}+19x+20}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339226352\">\n<div id=\"fs-id1167339226353\">\n<p id=\"fs-id1167339226354\">[latex]\\frac{{a}^{2}+9a+18}{{a}^{2}+3a-18}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339226415\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339226415\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339226415\"]\n<p id=\"fs-id1167339226416\">[latex]\\frac{a+3}{a-3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196454\">\n<div id=\"fs-id1167339196456\">\n<p id=\"fs-id1167339196457\">[latex]\\frac{3{c}^{2}+25c-18}{3{c}^{2}-23c+14}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339196524\">[latex]\\frac{12{n}^{2}-29n-8}{28{n}^{2}-5n-3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339260411\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339260411\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339260411\"]\n<p id=\"fs-id1167339260412\">[latex]\\frac{3n-8}{7n-3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339260450\">For the following exercises, multiply the rational expressions and express the product in simplest form.<\/p>\n\n<div id=\"fs-id1167339260454\">\n<div id=\"fs-id1167339260455\">\n<p id=\"fs-id1167339260456\">[latex]\\frac{{x}^{2}-x-6}{2{x}^{2}+x-6}\\cdot \\frac{2{x}^{2}+7x-15}{{x}^{2}-9}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339303584\">\n<div id=\"fs-id1167339303585\">\n<p id=\"fs-id1167339303586\">[latex]\\frac{{c}^{2}+2c-24}{{c}^{2}+12c+36}\\cdot \\frac{{c}^{2}-10c+24}{{c}^{2}-8c+16}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339303701\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339303701\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339303701\"]\n<p id=\"fs-id1167339303702\">[latex]\\frac{c-6}{c+6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339218644\">[latex]\\frac{2{d}^{2}+9d-35}{{d}^{2}+10d+21}\\cdot \\frac{3{d}^{2}+2d-21}{3{d}^{2}+14d-49}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339218766\">\n<div id=\"fs-id1167339218767\">\n<p id=\"fs-id1167339218768\">[latex]\\frac{10{h}^{2}-9h-9}{2{h}^{2}-19h+24}\\cdot \\frac{{h}^{2}-16h+64}{5{h}^{2}-37h-24}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167339239122\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339239122\"]\n<p id=\"fs-id1167339239122\">[latex]1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239131\">\n<div id=\"fs-id1167339239132\">\n<p id=\"fs-id1167339239133\">[latex]\\frac{6{b}^{2}+13b+6}{4{b}^{2}-9}\\cdot \\frac{6{b}^{2}+31b-30}{18{b}^{2}-3b-10}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339321412\">\n<div id=\"fs-id1167339321413\">\n<p id=\"fs-id1167339321414\">[latex]\\frac{2{d}^{2}+15d+25}{4{d}^{2}-25}\\cdot \\frac{2{d}^{2}-15d+25}{25{d}^{2}-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339344433\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339344433\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339344433\"]\n<p id=\"fs-id1167339344434\">[latex]\\frac{{d}^{2}-25}{25{d}^{2}-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339344484\">\n<div id=\"fs-id1167339344485\">\n<p id=\"fs-id1167339344486\">[latex]\\frac{6{x}^{2}-5x-50}{15{x}^{2}-44x-20}\\cdot \\frac{20{x}^{2}-7x-6}{2{x}^{2}+9x+10}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138825\">\n<div id=\"fs-id1167339138826\">\n<p id=\"fs-id1167339138827\">[latex]\\frac{{t}^{2}-1}{{t}^{2}+4t+3}\\cdot \\frac{{t}^{2}+2t-15}{{t}^{2}-4t+3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339138936\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339138936\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339138936\"]\n<p id=\"fs-id1167339138937\">[latex]\\frac{t+5}{t+3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273802\">\n<div id=\"fs-id1167339273803\">\n<p id=\"fs-id1167339273804\">[latex]\\frac{2{n}^{2}-n-15}{6{n}^{2}+13n-5}\\cdot \\frac{12{n}^{2}-13n+3}{4{n}^{2}-15n+9}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273926\">\n<div id=\"fs-id1167339273927\">\n<p id=\"fs-id1167339273928\">[latex]\\frac{36{x}^{2}-25}{6{x}^{2}+65x+50}\\cdot \\frac{3{x}^{2}+32x+20}{18{x}^{2}+27x+10}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339428060\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339428060\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339428060\"]\n<p id=\"fs-id1167339428061\">[latex]\\frac{6x-5}{6x+5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339428098\">For the following exercises, divide the rational expressions.<\/p>\n\n<div id=\"fs-id1167339428102\">\n<div id=\"fs-id1167339428103\">\n<p id=\"fs-id1167339428104\">[latex]\\frac{3{y}^{2}-7y-6}{2{y}^{2}-3y-9}\u00f7\\frac{{y}^{2}+y-2}{2{y}^{2}+y-3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339227755\">\n<div id=\"fs-id1167339227756\">\n<p id=\"fs-id1167339227757\">[latex]\\frac{6{p}^{2}+p-12}{8{p}^{2}+18p+9}\u00f7\\frac{6{p}^{2}-11p+4}{2{p}^{2}+11p-6}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339314058\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339314058\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339314058\"]\n<p id=\"fs-id1167339314059\">[latex]\\frac{p+6}{4p+3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339314094\">\n<div id=\"fs-id1167339314096\">\n<p id=\"fs-id1167339314097\">[latex]\\frac{{q}^{2}-9}{{q}^{2}+6q+9}\u00f7\\frac{{q}^{2}-2q-3}{{q}^{2}+2q-3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339314205\">\n<div id=\"fs-id1167339314206\">\n<p id=\"fs-id1167339314208\">[latex]\\frac{18{d}^{2}+77d-18}{27{d}^{2}-15d+2}\u00f7\\frac{3{d}^{2}+29d-44}{9{d}^{2}-15d+4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339214028\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339214028\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339214028\"]\n<p id=\"fs-id1167339214029\">[latex]\\frac{2d+9}{d+11}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214064\">\n<div id=\"fs-id1167339214065\">\n<p id=\"fs-id1167339214066\">[latex]\\frac{16{x}^{2}+18x-55}{32{x}^{2}-36x-11}\u00f7\\frac{2{x}^{2}+17x+30}{4{x}^{2}+25x+6}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339222916\">\n<div id=\"fs-id1167339222917\">\n<p id=\"fs-id1167339222918\">[latex]\\frac{144{b}^{2}-25}{72{b}^{2}-6b-10}\u00f7\\frac{18{b}^{2}-21b+5}{36{b}^{2}-18b-10}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"fs-id1167339222916\">\n<div>[reveal-answer q=\"659011\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"659011\"][latex]\\frac{12b+5}{3b-1}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339280911\">\n<div id=\"fs-id1167339280912\">\n<p id=\"fs-id1167339280914\">[latex]\\frac{16{a}^{2}-24a+9}{4{a}^{2}+17a-15}\u00f7\\frac{16{a}^{2}-9}{4{a}^{2}+11a+6}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339281031\">\n<div>\n<p id=\"fs-id1167339281033\">[latex]\\frac{22{y}^{2}+59y+10}{12{y}^{2}+28y-5}\u00f7\\frac{11{y}^{2}+46y+8}{24{y}^{2}-10y+1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339225605\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339225605\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339225605\"]\n<p id=\"fs-id1167339225606\">[latex]\\frac{4y-1}{y+4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339225642\">\n<div id=\"fs-id1167339225643\">\n<p id=\"fs-id1167339225644\">[latex]\\frac{9{x}^{2}+3x-20}{3{x}^{2}-7x+4}\u00f7\\frac{6{x}^{2}+4x-10}{{x}^{2}-2x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339225767\">For the following exercises, add and subtract the rational expressions, and then simplify.<\/p>\n\n<div id=\"fs-id1167339225771\">\n<div id=\"fs-id1167339225772\">\n<p id=\"fs-id1167339225774\">[latex]\\frac{4}{x}+\\frac{10}{y}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339317636\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339317636\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339317636\"]\n<p id=\"fs-id1167339317638\">[latex]\\frac{10x+4y}{xy}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317673\">\n<div>\n<p id=\"fs-id1167339317675\">[latex]\\frac{12}{2q}-\\frac{6}{3p}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317719\">\n<div id=\"fs-id1167339317720\">\n<p id=\"fs-id1167339317721\">[latex]\\frac{4}{a+1}+\\frac{5}{a-3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339317766\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339317766\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339317766\"]\n<p id=\"fs-id1167339317767\">[latex]\\frac{9a-7}{{a}^{2}-2a-3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317816\">\n<div id=\"fs-id1167339317817\">\n<p id=\"fs-id1167339317818\">[latex]\\frac{c+2}{3}-\\frac{c-4}{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317863\">\n<div id=\"fs-id1167339317864\">\n<p id=\"fs-id1167339317865\">[latex]\\frac{y+3}{y-2}+\\frac{y-3}{y+1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339315713\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339315713\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339315713\"]\n<p id=\"fs-id1167339315714\">[latex]\\frac{2{y}^{2}-y+9}{{y}^{2}-y-2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339315772\">\n<div id=\"fs-id1167339315773\">\n<p id=\"fs-id1167339315774\">[latex]\\frac{x-1}{x+1}-\\frac{2x+3}{2x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339315838\">\n<div id=\"fs-id1167339315840\">\n<p id=\"fs-id1167339315841\">[latex]\\frac{3z}{z+1}+\\frac{2z+5}{z-2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339315900\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339315900\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339315900\"]\n<p id=\"fs-id1167339315901\">[latex]\\frac{5{z}^{2}+z+5}{{z}^{2}-z-2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239217\">\n<div>[latex]\\frac{4p}{p+1}-\\frac{p+1}{4p}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339239275\">\n<div id=\"fs-id1167339239276\">\n<p id=\"fs-id1167339239277\">[latex]\\frac{x}{x+1}+\\frac{y}{y+1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339239322\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339239322\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339239322\"]\n<p id=\"fs-id1167339239323\">[latex]\\frac{x+2xy+y}{x+xy+y+1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339239375\">For the following exercises, simplify the rational expression.<\/p>\n\n<div id=\"fs-id1167339239378\">\n<div id=\"fs-id1167339239380\">\n<p id=\"fs-id1167339239381\">[latex]\\frac{\\frac{6}{y}-\\frac{4}{x}}{y}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239420\">\n<div id=\"fs-id1167339239421\">\n<p id=\"fs-id1167339239422\">[latex]\\frac{\\frac{2}{a}+\\frac{7}{b}}{b}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339437819\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339437819\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339437819\"]\n[latex]\\frac{2b+7a}{a{b}^{2}}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1167339437862\">\n<div id=\"fs-id1167339437863\">\n<p id=\"fs-id1167339437864\">[latex]\\frac{\\frac{x}{4}-\\frac{p}{8}}{p}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339437904\">\n<div id=\"fs-id1167339437905\">\n<p id=\"fs-id1167339437906\">[latex]\\frac{\\frac{3}{a}+\\frac{b}{6}}{\\frac{2b}{3a}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339437965\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339437965\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339437965\"]\n<p id=\"fs-id1167339437966\">[latex]\\frac{18+ab}{4b}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339438000\">\n<div id=\"fs-id1167339438001\">\n<p id=\"fs-id1167339438002\">[latex]\\frac{\\frac{3}{x+1}+\\frac{2}{x-1}}{\\frac{x-1}{x+1}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339438081\">\n<div id=\"fs-id1167339240839\">\n<p id=\"fs-id1167339240840\">[latex]\\frac{\\frac{a}{b}-\\frac{b}{a}}{\\frac{a+b}{ab}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339240902\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339240902\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339240902\"]\n<p id=\"fs-id1167339240903\">[latex]a-b[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339240919\">\n<div id=\"fs-id1167339240920\">\n<p id=\"fs-id1167339240921\">[latex]\\frac{\\frac{2x}{3}+\\frac{4x}{7}}{\\frac{x}{2}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339240980\">\n<div id=\"fs-id1167339240982\">\n<p id=\"fs-id1167339240983\">[latex]\\frac{\\frac{2c}{c+2}+\\frac{c-1}{c+1}}{\\frac{2c+1}{c+1}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339241076\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339241076\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339241076\"]\n<p id=\"fs-id1167339241078\">[latex]\\frac{3{c}^{2}+3c-2}{2{c}^{2}+5c+2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339433020\">[latex]\\frac{\\frac{x}{y}-\\frac{y}{x}}{\\frac{x}{y}+\\frac{y}{x}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433082\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1167339433087\">\n<div id=\"fs-id1167339433088\">\n<p id=\"fs-id1167339433089\">Brenda is placing tile on her bathroom floor. The area of the floor is[latex]\\,15{x}^{2}-8x-7\\,[\/latex]ft<sup>2<\/sup>. The area of one tile is[latex]\\,{x}^{2}-2x+1{\\text{ft}}^{2}.\\,[\/latex]To find the number of tiles needed, simplify the rational expression:[latex]\\,\\frac{15{x}^{2}-8x-7}{{x}^{2}-2x+1}.[\/latex]<\/p>\n<span id=\"fs-id1167339433096\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132210\/CNX_CAT_Figure_01_06_201.jpg\" alt=\"A rectangle that\u2019s labeled: Area = fifteen times x squared minus eight times x minus seven.\"><\/span>\n\n<\/div>\n<div id=\"fs-id1167339433108\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339433108\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339433108\"]\n<p id=\"fs-id1167339433109\">[latex]\\frac{15x+7}{x-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433146\">\n<div id=\"fs-id1167339433147\">\n<p id=\"fs-id1167339433148\">The area of Sandy\u2019s yard is[latex]\\,25{x}^{2}-625\\,[\/latex]ft<sup>2<\/sup>. A patch of sod has an area of[latex]\\,{x}^{2}-10x+25\\,[\/latex]ft<sup>2<\/sup>. Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433218\">\n<div id=\"fs-id1167339433219\">\n<p id=\"fs-id1167339433220\">Aaron wants to mulch his garden. His garden is[latex]\\,{x}^{2}+18x+81\\,[\/latex]ft<sup>2<\/sup>. One bag of mulch covers[latex]\\,{x}^{2}-81\\,[\/latex]ft<sup>2<\/sup>. Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.<\/p>\n\n<\/div>\n<div id=\"fs-id1167339433288\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339433288\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339433288\"]\n<p id=\"fs-id1167339433289\">[latex]\\frac{x+9}{x-9}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299394\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1167339299400\">For the following exercises, perform the given operations and simplify.<\/p>\n\n<div id=\"fs-id1167339299403\">\n<div id=\"fs-id1167339299404\">\n<p id=\"fs-id1167339299405\">[latex]\\frac{{x}^{2}+x-6}{{x}^{2}-2x-3}\\cdot \\frac{2{x}^{2}-3x-9}{{x}^{2}-x-2}\u00f7\\frac{10{x}^{2}+27x+18}{{x}^{2}+2x+1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299574\">\n<div id=\"fs-id1167339299576\">\n<p id=\"fs-id1167339299577\">[latex]\\frac{\\frac{3{y}^{2}-10y+3}{3{y}^{2}+5y-2}\\cdot \\frac{2{y}^{2}-3y-20}{2{y}^{2}-y-15}}{y-4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339299716\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339299716\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339299716\"]\n<p id=\"fs-id1167339299717\">[latex]\\frac{1}{y+2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219221\">\n<div id=\"fs-id1167339219222\">\n<p id=\"fs-id1167339219223\">[latex]\\frac{\\frac{4a+1}{2a-3}+\\frac{2a-3}{2a+3}}{\\frac{4{a}^{2}+9}{a}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219330\">\n<div id=\"fs-id1167339219331\">\n<p id=\"fs-id1167339219332\">[latex]\\frac{{x}^{2}+7x+12}{{x}^{2}+x-6}\u00f7\\frac{3{x}^{2}+19x+28}{8{x}^{2}-4x-24}\u00f7\\frac{2{x}^{2}+x-3}{3{x}^{2}+4x-7}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339219508\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339219508\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339219508\"]\n<p id=\"fs-id1167339219509\">[latex]4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219520\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1167339219527\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/d0a86917-6447-4b49-bed1-efb5c352cc77\">Real Numbers: Algebra Essentials<\/a><\/h4>\n<p id=\"fs-id1167339219537\">For the following exercises, perform the given operations.<\/p>\n\n<div id=\"fs-id1167339219540\">\n<div id=\"fs-id1167339331066\">\n<p id=\"fs-id1167339331067\">[latex]{\\left(5-3\\cdot 2\\right)}^{2}-6[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339331113\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339331113\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339331113\"]\n<p id=\"fs-id1167339331114\">[latex]-5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331128\">\n<div id=\"fs-id1167339331129\">\n<p id=\"fs-id1167339331130\">[latex]64\u00f7\\left(2\\cdot 8\\right)+14\u00f77[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331169\">\n<div id=\"fs-id1167339331170\">\n<p id=\"fs-id1167339331171\">[latex]2\\cdot {5}^{2}+6\u00f72[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339331203\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339331203\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339331203\"]\n<p id=\"fs-id1167339331204\">53<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339331208\">For the following exercises, solve the equation.<\/p>\n\n<div id=\"fs-id1167339331211\">\n<div id=\"fs-id1167339331212\">\n<p id=\"fs-id1167339331213\">[latex]5x+9=-11[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331237\">\n<div id=\"fs-id1167339331238\">\n<p id=\"fs-id1167339331240\">[latex]2y+{4}^{2}=64[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339331269\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339331269\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339331269\"]\n<p id=\"fs-id1167339331270\">[latex]y=24[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339331286\">For the following exercises, simplify the expression.<\/p>\n\n<div id=\"fs-id1167339331290\">\n<div id=\"fs-id1167339331291\">\n<p id=\"fs-id1167339331292\">[latex]9\\left(y+2\\right)\u00f73\\cdot 2+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331333\">\n<div id=\"fs-id1167339331334\">\n<p id=\"fs-id1167339331336\">[latex]3m\\left(4+7\\right)-m[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339331371\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339331371\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339331371\"]\n<p id=\"fs-id1167339331372\">[latex]32m[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339331386\">For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.<\/p>\n\n<div id=\"fs-id1167339331390\">\n<div id=\"fs-id1167339331391\">\n<p id=\"fs-id1167339331392\">11<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331395\">\n<div id=\"fs-id1167339331396\">\n<p id=\"fs-id1167339331398\">0<\/p>\n\n<\/div>\n<div id=\"fs-id1167339331401\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339331401\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339331401\"]\n<p id=\"fs-id1167339331402\">whole<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331405\">\n<div id=\"fs-id1167339331406\">\n<p id=\"fs-id1167339331407\">[latex]\\frac{5}{6}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432116\">\n<div id=\"fs-id1167339432117\">\n<p id=\"fs-id1167339432118\">[latex]\\sqrt{11}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339432136\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339432136\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339432136\"]\n<p id=\"fs-id1167339432137\">irrational<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432142\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/065c3182-adac-48fc-a3ab-03d487982dd5\">Exponents and Scientific Notation<\/a><\/h4>\n<p id=\"fs-id1167339432152\">For the following exercises, simplify the expression.<\/p>\n\n<div id=\"fs-id1167339432155\">\n<div id=\"fs-id1167339432156\">\n<p id=\"fs-id1167339432157\">[latex]{2}^{2}\\cdot {2}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432187\">\n<div id=\"fs-id1167339432188\">\n<p id=\"fs-id1167339432189\">[latex]\\frac{{4}^{5}}{{4}^{3}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339432227\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339432227\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339432227\"]\n<p id=\"fs-id1167339432228\">[latex]16[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432240\">\n<div id=\"fs-id1167339432241\">\n<p id=\"fs-id1167339432242\">[latex]{\\left(\\frac{{a}^{2}}{{b}^{3}}\\right)}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432299\">\n<div id=\"fs-id1167339432300\">\n<p id=\"fs-id1167339432301\">[latex]\\frac{6{a}^{2}\\cdot {a}^{0}}{2{a}^{-4}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339432360\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339432360\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339432360\"]\n<p id=\"fs-id1167339432361\">[latex]{a}^{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432379\">\n<div id=\"fs-id1167339432380\">\n<p id=\"fs-id1167339432381\">[latex]\\frac{{\\left(xy\\right)}^{4}}{{y}^{3}}\\cdot \\frac{2}{{x}^{5}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432456\">\n<div id=\"fs-id1167339432457\">\n<p id=\"fs-id1167339432458\">[latex]\\frac{{4}^{-2}{x}^{3}{y}^{-3}}{2{x}^{0}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339295802\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339295802\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339295802\"]\n<p id=\"fs-id1167339295803\">[latex]\\frac{{x}^{3}}{32{y}^{3}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339295842\">\n<div id=\"fs-id1167339295843\">\n<p id=\"fs-id1167339295844\">[latex]{\\left(\\frac{2{x}^{2}}{y}\\right)}^{-2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339295900\">\n<div id=\"fs-id1167339295901\">\n<p id=\"fs-id1167339295902\">[latex]\\left(\\frac{16{a}^{3}}{{b}^{2}}\\right){\\left(4a{b}^{-1}\\right)}^{-2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339295996\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339295996\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339295996\"]\n<p id=\"fs-id1167339295997\">[latex]a[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296005\">\n<div id=\"fs-id1167339296006\">\n<p id=\"fs-id1167339296007\">Write the number in standard notation:[latex]\\,2.1314\\,\u00d7\\,{10}^{-6}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296042\">\n<div id=\"fs-id1167339296043\">\n<p id=\"fs-id1167339296044\">Write the number in scientific notation: 16,340,000<\/p>\n\n<\/div>\n<div id=\"fs-id1167339296047\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339296047\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339296047\"]\n<p id=\"fs-id1167339296048\">[latex]1.634\\,\u00d7\\,{10}^{7}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296075\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/1834bc64-7094-4df1-a530-b3166a295697\">Radicals and Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167339296085\">For the following exercises, find the principal square root.<\/p>\n\n<div id=\"fs-id1167339296088\">\n<div id=\"fs-id1167339296089\">\n<p id=\"fs-id1167339296090\">[latex]\\sqrt{121}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296108\">\n<div id=\"fs-id1167339296110\">\n<p id=\"fs-id1167339296111\">[latex]\\sqrt{196}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339296129\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339296129\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339296129\"]\n<p id=\"fs-id1167339296130\">14<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296133\">\n<div id=\"fs-id1167339296134\">\n<p id=\"fs-id1167339296135\">[latex]\\sqrt{361}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296153\">\n<div id=\"fs-id1167339296154\">\n<p id=\"fs-id1167339296155\">[latex]\\sqrt{75}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339296174\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339296174\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339296174\"]\n<p id=\"fs-id1167339296175\">[latex]5\\sqrt{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296192\">\n<div id=\"fs-id1167339296193\">\n<p id=\"fs-id1167339296194\">[latex]\\sqrt{162}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296212\">\n<div id=\"fs-id1167339296213\">\n<p id=\"fs-id1167339296214\">[latex]\\sqrt{\\frac{32}{25}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339296245\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339296245\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339296245\"]\n<p id=\"fs-id1167339296246\">[latex]\\frac{4\\sqrt{2}}{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206368\">\n<div id=\"fs-id1167339206369\">\n<p id=\"fs-id1167339206370\">[latex]\\sqrt{\\frac{80}{81}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206401\">\n<div id=\"fs-id1167339206402\">\n<p id=\"fs-id1167339206403\">[latex]\\sqrt{\\frac{49}{1250}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339206434\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339206434\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339206434\"]\n<p id=\"fs-id1167339206435\">[latex]\\frac{7\\sqrt{2}}{50}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206465\">\n<div id=\"fs-id1167339206466\">\n<p id=\"fs-id1167339206467\">[latex]\\frac{2}{4+\\sqrt{2}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206496\">\n<div id=\"fs-id1167339206497\">\n<p id=\"fs-id1167339206498\">[latex]4\\sqrt{3}+6\\sqrt{3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339206525\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339206525\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339206525\"]\n<p id=\"fs-id1167339206526\">[latex]10\\sqrt{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206544\">\n<div id=\"fs-id1167339206545\">\n<p id=\"fs-id1167339206546\">[latex]12\\sqrt{5}-13\\sqrt{5}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206573\">\n<div id=\"fs-id1167339206574\">\n<p id=\"fs-id1167339206576\">[latex]\\sqrt[5]{-243}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339206599\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339206599\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339206599\"]\n<p id=\"fs-id1167339206600\">[latex]-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206614\">\n<div id=\"fs-id1167339206615\">\n<p id=\"fs-id1167339206616\">[latex]\\frac{\\sqrt[3]{250}}{\\sqrt[3]{-8}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206663\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f7978ad8-ed27-4fe4-8a19-26a031ba97ad\">Polynomials<\/a><\/h4>\n<p id=\"fs-id1167339206673\">For the following exercises, perform the given operations and simplify.<\/p>\n\n<div id=\"fs-id1167339206677\">\n<div id=\"fs-id1167339206678\">\n<p id=\"fs-id1167339206679\">[latex]\\left(3{x}^{3}+2x-1\\right)+\\left(4{x}^{2}-2x+7\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339206757\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339206757\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339206757\"]\n<p id=\"fs-id1167339206758\">[latex]3{x}^{3}+4{x}^{2}+6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206797\">\n<div id=\"fs-id1167339206798\">\n<p id=\"fs-id1167339206799\">[latex]\\left(2y+1\\right)-\\left(2{y}^{2}-2y-5\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206863\">\n<div id=\"fs-id1167339206864\">\n<p id=\"fs-id1167339206865\">[latex]\\left(2{x}^{2}+3x-6\\right)+\\left(3{x}^{2}-4x+9\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339297739\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339297739\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339297739\"]\n<p id=\"fs-id1167339297740\">[latex]5{x}^{2}-x+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297770\">\n<div id=\"fs-id1167339297771\">\n<p id=\"fs-id1167339297772\">[latex]\\left(6{a}^{2}+3a+10\\right)-\\left(6{a}^{2}-3a+5\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297850\">\n<div id=\"fs-id1167339297851\">\n<p id=\"fs-id1167339297852\">[latex]\\left(k+3\\right)\\left(k-6\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339297896\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339297896\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339297896\"]\n<p id=\"fs-id1167339297897\">[latex]{k}^{2}-3k-18[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297927\">\n<div id=\"fs-id1167339297928\">\n<p id=\"fs-id1167339297929\">[latex]\\left(2h+1\\right)\\left(3h-2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297977\">\n<div id=\"fs-id1167339297978\">\n<p id=\"fs-id1167339297979\">[latex]\\left(x+1\\right)\\left({x}^{2}+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339298030\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339298030\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339298030\"]\n<p id=\"fs-id1167339298032\">[latex]{x}^{3}+{x}^{2}+x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298071\">\n<div id=\"fs-id1167339298072\">\n<p id=\"fs-id1167339298073\">[latex]\\left(m-2\\right)\\left({m}^{2}+2m-3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298131\">\n<div id=\"fs-id1167339298132\">\n<p id=\"fs-id1167339298133\">[latex]\\left(a+2b\\right)\\left(3a-b\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339298181\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339298181\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339298181\"]\n<p id=\"fs-id1167339298182\">[latex]3{a}^{2}+5ab-2{b}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298225\">\n<div id=\"fs-id1167339298226\">\n<p id=\"fs-id1167339298227\">[latex]\\left(x+y\\right)\\left(x-y\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298272\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/49cf2d69-1d37-49aa-9e61-16da4c52ce37\">Factoring Polynomials<\/a><\/h4>\n<p id=\"fs-id1167339298282\">For the following exercises, find the greatest common factor.<\/p>\n\n<div id=\"fs-id1167339298285\">\n<div id=\"fs-id1167339298286\">\n<p id=\"fs-id1167339298287\">[latex]81p+9pq-27{p}^{2}{q}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339298332\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339298332\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339298332\"]\n<p id=\"fs-id1167339298333\">[latex]9p[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308685\">\n<div id=\"fs-id1167339308686\">\n<p id=\"fs-id1167339308687\">[latex]12{x}^{2}y+4x{y}^{2}-18xy[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308735\">\n<div id=\"fs-id1167339308736\">\n<p id=\"fs-id1167339308737\">[latex]88{a}^{3}b+4{a}^{2}b-144{a}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339308789\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339308789\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339308789\"]\n<p id=\"fs-id1167339308790\">[latex]4{a}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339308811\">For the following exercises, factor the polynomial.<\/p>\n\n<div id=\"fs-id1167339308814\">\n<div id=\"fs-id1167339308815\">\n<p id=\"fs-id1167339308816\">[latex]2{x}^{2}-9x-18[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308848\">\n<div id=\"fs-id1167339308849\">\n<p id=\"fs-id1167339308850\">[latex]8{a}^{2}+30a-27[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339308882\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339308882\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339308882\"]\n<p id=\"fs-id1167339308883\">[latex]\\left(4a-3\\right)\\left(2a+9\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308931\">\n<div id=\"fs-id1167339308932\">\n<p id=\"fs-id1167339308933\">[latex]{d}^{2}-5d-66[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308963\">\n<div id=\"fs-id1167339308964\">\n<p id=\"fs-id1167339308965\">[latex]{x}^{2}+10x+25[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339308995\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339308995\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339308995\"]\n<p id=\"fs-id1167339308996\">[latex]{\\left(x+5\\right)}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309032\">\n<div id=\"fs-id1167339309034\">\n<p id=\"fs-id1167339309035\">[latex]{y}^{2}-6y+9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309064\">\n<div id=\"fs-id1167339309066\">\n<p id=\"fs-id1167339309067\">[latex]4{h}^{2}-12hk+9{k}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339309109\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339309109\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339309109\"]\n<p id=\"fs-id1167339309110\">[latex]{\\left(2h-3k\\right)}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309151\">\n<div id=\"fs-id1167339309152\">\n<p id=\"fs-id1167339309153\">[latex]361{x}^{2}-121[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309179\">\n<div id=\"fs-id1167339309180\">\n<p id=\"fs-id1167339309181\">[latex]{p}^{3}+216[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339309204\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339309204\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339309204\"]\n<p id=\"fs-id1167339309205\">[latex]\\left(p+6\\right)\\left({p}^{2}-6p+36\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309263\">\n<div id=\"fs-id1167339309264\">\n<p id=\"fs-id1167339309265\">[latex]8{x}^{3}-125[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309291\">\n<div id=\"fs-id1167339309292\">\n<p id=\"fs-id1167339309293\">[latex]64{q}^{3}-27{p}^{3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339309327\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339309327\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339309327\"]\n<p id=\"fs-id1167339309328\">[latex]\\left(4q-3p\\right)\\left(16{q}^{2}+12pq+9{p}^{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309403\">\n<div id=\"fs-id1167339309404\">\n<p id=\"fs-id1167339309405\">[latex]4x{\\left(x-1\\right)}^{-\\frac{1}{4}}+3{\\left(x-1\\right)}^{\\frac{3}{4}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309499\">\n<div id=\"fs-id1167339309500\">\n<p id=\"fs-id1167339309501\">[latex]3p{\\left(p+3\\right)}^{\\frac{1}{3}}-8{\\left(p+3\\right)}^{\\frac{4}{3}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339433736\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339433736\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339433736\"]\n<p id=\"fs-id1167339433737\">[latex]{\\left(p+3\\right)}^{\\frac{1}{3}}\\left(-5p-24\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433805\">\n<div id=\"fs-id1167339433806\">\n<p id=\"fs-id1167339433808\">[latex]4r{\\left(2r-1\\right)}^{-\\frac{2}{3}}-5{\\left(2r-1\\right)}^{\\frac{1}{3}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433907\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/45953190-e727-41e0-9e6a-834e092ee148\">Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167339433916\">For the following exercises, simplify the expression.<\/p>\n\n<div id=\"fs-id1167339433919\">\n<div id=\"fs-id1167339433920\">\n<p id=\"fs-id1167339433922\">[latex]\\frac{{x}^{2}-x-12}{{x}^{2}-8x+16}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339433980\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339433980\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339433980\"]\n<p id=\"fs-id1167339433981\">[latex]\\frac{x+3}{x-4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434014\">\n<div id=\"fs-id1167339434015\">\n<p id=\"fs-id1167339434016\">[latex]\\frac{4{y}^{2}-25}{4{y}^{2}-20y+25}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434075\">\n<div id=\"fs-id1167339434076\">\n<p id=\"fs-id1167339434077\">[latex]\\frac{2{a}^{2}-a-3}{2{a}^{2}-6a-8}\\cdot \\frac{5{a}^{2}-19a-4}{10{a}^{2}-13a-3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339434201\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339434201\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339434201\"]\n<p id=\"fs-id1167339434202\">[latex]\\frac{1}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434220\">\n<div id=\"fs-id1167339434221\">\n<p id=\"fs-id1167339434222\">[latex]\\frac{d-4}{{d}^{2}-9}\\cdot \\frac{d-3}{{d}^{2}-16}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434297\">\n<div id=\"fs-id1167339434298\">\n<p id=\"fs-id1167339434299\">[latex]\\frac{{m}^{2}+5m+6}{2{m}^{2}-5m-3}\u00f7\\frac{2{m}^{2}+3m-9}{4{m}^{2}-4m-3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339434421\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339434421\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339434421\"]\n<p id=\"fs-id1167339434422\">[latex]\\frac{m+2}{m-3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434455\">\n<div id=\"fs-id1167339434456\">\n<p id=\"fs-id1167339434457\">[latex]\\frac{4{d}^{2}-7d-2}{6{d}^{2}-17d+10}\u00f7\\frac{8{d}^{2}+6d+1}{6{d}^{2}+7d-10}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434581\">\n<div id=\"fs-id1167339434582\">\n<p id=\"fs-id1167339434583\">[latex]\\frac{10}{x}+\\frac{6}{y}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339434616\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339434616\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339434616\"]\n<p id=\"fs-id1167339434617\">[latex]\\frac{6x+10y}{xy}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434652\">\n<div id=\"fs-id1167339434653\">\n<p id=\"fs-id1167339434654\">[latex]\\frac{12}{{a}^{2}+2a+1}-\\frac{3}{{a}^{2}-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434724\">\n<div id=\"fs-id1167339434725\">\n<p id=\"fs-id1167339434726\">[latex]\\frac{\\frac{1}{d}+\\frac{2}{c}}{\\frac{6c+12d}{dc}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339434792\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339434792\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339434792\"]\n<p id=\"fs-id1167339434793\">[latex]\\frac{1}{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434811\">\n<div id=\"fs-id1167339434812\">\n<p id=\"fs-id1167339434813\">[latex]\\frac{\\frac{3}{x}-\\frac{7}{y}}{\\frac{2}{x}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434864\" class=\"practice-test\">\n<h3>Chapter Practice Test<\/h3>\n<p id=\"fs-id1167339434872\">For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.<\/p>\n\n<div id=\"fs-id1167339434876\">\n<div id=\"fs-id1167339434877\">\n<p id=\"fs-id1167339434878\">[latex]-13[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339434892\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339434892\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339434892\"]\n<p id=\"fs-id1167339434893\">rational<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434896\">\n<div id=\"fs-id1167339434897\">\n<p id=\"fs-id1167339434898\">[latex]\\sqrt{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339434913\">For the following exercises, evaluate the equations.<\/p>\n\n<div id=\"fs-id1167339434917\">\n<div id=\"fs-id1167339434918\">\n<p id=\"fs-id1167339434919\">[latex]2\\left(x+3\\right)-12=18[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339434956\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339434956\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339434956\"]\n<p id=\"fs-id1167339434957\">[latex]x=12[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434973\">\n<div id=\"fs-id1167339434974\">\n<p id=\"fs-id1167339434975\">[latex]y{\\left(3+3\\right)}^{2}-26=10[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1167339435024\">\n<p id=\"fs-id1167339435025\">Write the number in standard notation:[latex]3.1415\\,\u00d7\\,{10}^{6}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339435052\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339435052\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339435052\"]\n<p id=\"fs-id1167339435053\">3,141,500<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342169\">\n<div id=\"fs-id1167339342170\">\n<p id=\"fs-id1167339342171\">Write the number in scientific notation: 0.0000000212.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339342174\">For the following exercises, simplify the expression.<\/p>\n\n<div id=\"fs-id1167339342178\">\n<div id=\"fs-id1167339342179\">\n<p id=\"fs-id1167339342180\">[latex]-2\\cdot {\\left(2+3\\cdot 2\\right)}^{2}+144[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342232\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342232\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342232\"]\n<p id=\"fs-id1167339342233\">[latex]16[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342245\">\n<div id=\"fs-id1167339342246\">\n<p id=\"fs-id1167339342247\">[latex]4\\left(x+3\\right)-\\left(6x+2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342297\">\n<div id=\"fs-id1167339342298\">\n<p id=\"fs-id1167339342299\">[latex]{3}^{5}\\cdot {3}^{-3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342334\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342334\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342334\"]\n<p id=\"fs-id1167339342336\">9<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342339\">\n<div id=\"fs-id1167339342340\">\n<p id=\"fs-id1167339342341\">[latex]{\\left(\\frac{2}{3}\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342381\">\n<div id=\"fs-id1167339342382\">\n<p id=\"fs-id1167339342383\">[latex]\\frac{8{x}^{3}}{{\\left(2x\\right)}^{2}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342441\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342441\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342441\"]\n<p id=\"fs-id1167339342442\">[latex]2x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342457\">\n<div id=\"fs-id1167339342458\">\n<p id=\"fs-id1167339342459\">[latex]\\left(16{y}^{0}\\right)2{y}^{-2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342506\">\n<div id=\"fs-id1167339342507\">\n<p id=\"fs-id1167339342508\">[latex]\\sqrt{441}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342526\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342526\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342526\"]\n<p id=\"fs-id1167339342527\">21<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342530\">\n<div id=\"fs-id1167339342532\">\n<p id=\"fs-id1167339342533\">[latex]\\sqrt{490}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342551\">\n<div id=\"fs-id1167339342552\">\n<p id=\"fs-id1167339342553\">[latex]\\sqrt{\\frac{9x}{16}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342586\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342586\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342586\"]\n<p id=\"fs-id1167339342587\">[latex]\\frac{3\\sqrt{x}}{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342614\">\n<div id=\"fs-id1167339342615\">\n<p id=\"fs-id1167339342616\">[latex]\\frac{\\sqrt{121{b}^{2}}}{1+\\sqrt{b}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342663\">\n<div id=\"fs-id1167339342664\">\n<p id=\"fs-id1167339342665\">[latex]6\\sqrt{24}+7\\sqrt{54}-12\\sqrt{6}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342710\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342710\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342710\"]\n<p id=\"fs-id1167339342711\">[latex]21\\sqrt{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342728\">\n<div id=\"fs-id1167339342729\">\n<p id=\"fs-id1167339342730\">[latex]\\frac{\\sqrt[3]{-8}}{\\sqrt[4]{625}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342776\">\n<div id=\"fs-id1167339342777\">\n<p id=\"fs-id1167339342778\">[latex]\\left(13{q}^{3}+2{q}^{2}-3\\right)-\\left(6{q}^{2}+5q-3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339342863\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339342863\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339342863\"]\n<p id=\"fs-id1167339342864\">[latex]13{q}^{3}-4{q}^{2}-5q[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342906\">\n<div id=\"fs-id1167339342907\">\n<p id=\"fs-id1167339342908\">[latex]\\left(6{p}^{2}+2p+1\\right)+\\left(9{p}^{2}-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342980\">\n<div id=\"fs-id1167339342981\">\n<p id=\"fs-id1167339342982\">[latex]\\left(n-2\\right)\\left({n}^{2}-4n+4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339343039\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339343039\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339343039\"]\n<p id=\"fs-id1167339343040\">[latex]{n}^{3}-6{n}^{2}+12n-8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343084\">\n<div id=\"fs-id1167339343085\">\n<p id=\"fs-id1167339343086\">[latex]\\left(a-2b\\right)\\left(2a+b\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339343134\">For the following exercises, factor the polynomial.<\/p>\n\n<div id=\"fs-id1167339343137\">\n<div id=\"fs-id1167339343138\">\n<p id=\"fs-id1167339343140\">[latex]16{x}^{2}-81[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339343165\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339343165\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339343165\"]\n<p id=\"fs-id1167339343166\">[latex]\\left(4x+9\\right)\\left(4x-9\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343214\">\n<div id=\"fs-id1167339343215\">\n<p id=\"fs-id1167339343216\">[latex]{y}^{2}+12y+36[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343246\">\n<div id=\"fs-id1167339343247\">\n<p id=\"fs-id1167339343248\">[latex]27{c}^{3}-1331[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339343274\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339343274\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339343274\"]\n<p id=\"fs-id1167339343275\">[latex]\\left(3c-11\\right)\\left(9{c}^{2}+33c+121\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343337\">\n<div id=\"fs-id1167339343338\">\n<p id=\"fs-id1167339343339\">[latex]3x{\\left(x-6\\right)}^{-\\frac{1}{4}}+2{\\left(x-6\\right)}^{\\frac{3}{4}}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339343433\">For the following exercises, simplify the expression.<\/p>\n\n<div id=\"fs-id1167339343436\">\n<div id=\"fs-id1167339343437\">\n<p id=\"fs-id1167339343438\">[latex]\\frac{2{z}^{2}+7z+3}{{z}^{2}-9}\\cdot \\frac{4{z}^{2}-15z+9}{4{z}^{2}-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339343547\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339343547\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339343547\"]\n<p id=\"fs-id1167339343548\">[latex]\\frac{4z-3}{2z-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343585\">\n<div id=\"fs-id1167339343586\">\n<p id=\"fs-id1167339343588\">[latex]\\frac{x}{y}+\\frac{2}{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343617\">\n<div id=\"fs-id1167339343618\">\n<p id=\"fs-id1167339343620\">[latex]\\frac{\\frac{a}{2b}-\\frac{2b}{9a}}{\\frac{3a-2b}{6a}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339343702\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339343702\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339343702\"]\n<p id=\"fs-id1167339343703\">[latex]\\frac{3a+2b}{3b}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1167339343742\">\n \t<dt>least common denominator<\/dt>\n \t<dd id=\"fs-id1167339343745\">the smallest multiple that two denominators have in common<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339343749\">\n \t<dt>rational expression<\/dt>\n \t<dd id=\"fs-id1167339343752\">the quotient of two polynomial expressions<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section students will:<\/p>\n<ul>\n<li>Simplify rational expressions.<\/li>\n<li>Multiply rational expressions.<\/li>\n<li>Divide rational expressions.<\/li>\n<li>Add and subtract rational expressions.<\/li>\n<li>Simplify complex rational expressions.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167339311757\">A pastry shop has fixed costs of[latex]\\,\\text{\\$}280\\,[\/latex]per week and variable costs of[latex]\\,\\text{\\$}9\\,[\/latex]per box of pastries. The shop\u2019s costs per week in terms of[latex]\\,x,[\/latex]the number of boxes made, is[latex]\\,280+9x.\\,[\/latex]We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.<\/p>\n<div id=\"fs-id1167339295641\" class=\"unnumbered aligncenter\">[latex]\\frac{280+9x}{x}[\/latex]<\/div>\n<p id=\"fs-id1167339261982\">Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.<\/p>\n<div id=\"fs-id1167339317968\" class=\"bc-section section\">\n<h3>Simplifying Rational Expressions<\/h3>\n<p id=\"fs-id1167339185999\">The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let\u2019s start with the rational expression shown.<\/p>\n<div id=\"fs-id1167339433455\" class=\"unnumbered aligncenter\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/div>\n<p id=\"fs-id1167339344788\">We can factor the numerator and denominator to rewrite the expression.<\/p>\n<div id=\"fs-id1167339429080\" class=\"unnumbered aligncenter\">[latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/div>\n<p id=\"fs-id1167339199768\">Then we can simplify that expression by canceling the common factor[latex]\\,\\left(x+4\\right).[\/latex]<\/p>\n<div id=\"fs-id1167339318196\" class=\"unnumbered aligncenter\">[latex]\\frac{x+4}{x+7}[\/latex]<\/div>\n<div id=\"fs-id1167339281673\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339240476\"><strong>Given a rational expression, simplify it.<\/strong><\/p>\n<ol id=\"fs-id1167339306312\" type=\"1\">\n<li>Factor the numerator and denominator.<\/li>\n<li>Cancel any common factors.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1167339303419\">\n<div id=\"fs-id1167339321615\">\n<h3>Simplifying Rational Expressions<\/h3>\n<p id=\"fs-id1167339197034\">Simplify[latex]\\,\\frac{{x}^{2}-9}{{x}^{2}+4x+3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339232113\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1167339155326\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lllll}\\frac{\\left(x+3\\right)\\left(x-3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x-3}{x+1}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331535\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339432914\">We can cancel the common factor because any expression divided by itself is equal to 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339324584\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339231124\"><strong>Can the[latex]\\,{x}^{2}\\,[\/latex]term be cancelled in <a class=\"autogenerated-content\" href=\"#Example_01_06_01\">(Figure)<\/a>?<\/strong><\/p>\n<p id=\"fs-id1167339281697\"><em>No. A factor is an expression that is multiplied by another expression. The[latex]\\,{x}^{2}\\,[\/latex]term is not a factor of the numerator or the denominator.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1167339154122\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_01\">\n<div id=\"fs-id1167339299924\">\n<p id=\"fs-id1167339299925\">Simplify[latex]\\,\\frac{x-6}{{x}^{2}-36}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339126332\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339166294\">[latex]\\frac{1}{x+6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339165929\" class=\"bc-section section\">\n<h3>Multiplying Rational Expressions<\/h3>\n<p id=\"fs-id1167339149550\">Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.<\/p>\n<div id=\"fs-id1167339185386\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339156547\"><strong>Given two rational expressions, multiply them.<\/strong><\/p>\n<ol id=\"fs-id1167339344628\" type=\"1\">\n<li>Factor the numerator and denominator.<\/li>\n<li>Multiply the numerators.<\/li>\n<li>Multiply the denominators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_02\" class=\"textbox examples\">\n<div id=\"fs-id1167339330081\">\n<div id=\"fs-id1167339296544\">\n<h3>Multiplying Rational Expressions<\/h3>\n<p id=\"fs-id1167339281353\">Multiply the rational expressions and show the product in simplest form:<\/p>\n<div id=\"fs-id1167339432908\" class=\"unnumbered aligncenter\">[latex]\\frac{\\left(x+5\\right)\\left(x-1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x-1\\right)}{\\left(x+5\\right)}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339228244\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1167339184717\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{lllll}\\frac{\\left(x+5\\right)\\left(x-1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x-1\\right)}{\\left(x+5\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Factor the numerator and denominator}.\\hfill \\\\ \\frac{\\left(x+5\\right)\\left(x-1\\right)\\left(2x-1\\right)}{3\\left(x+6\\right)\\left(x+5\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Multiply numerators and denominators}.\\hfill \\\\ \\frac{\\overline{)\\left(x+5\\right)}\\left(x-1\\right)\\left(2x-1\\right)}{3\\left(x+6\\right)\\overline{)\\left(x+5\\right)}}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Cancel common factors to simplify}.\\hfill \\\\ \\frac{\\left(x-1\\right)\\left(2x-1\\right)}{3\\left(x+6\\right)} \\hfill & \\hfill & \\hfill & \\hfill & \\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339281579\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_02\">\n<div id=\"fs-id1167339228502\">\n<p id=\"fs-id1167339228503\">Multiply the rational expressions and show the product in simplest form:<\/p>\n<div id=\"fs-id1167339228507\" class=\"unnumbered aligncenter\">[latex]\\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339228097\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339212908\">[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339281343\" class=\"bc-section section\">\n<h3>Dividing Rational Expressions<\/h3>\n<p id=\"fs-id1167339185206\">Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite[latex]\\,\\frac{1}{x}\u00f7\\frac{{x}^{2}}{3}\\,[\/latex]as the product[latex]\\,\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}.\\,[\/latex]Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.<\/p>\n<div id=\"fs-id1167339344761\" class=\"unnumbered aligncenter\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\n<div id=\"fs-id1167339117616\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339344893\"><strong>Given two rational expressions, divide them.<\/strong><\/p>\n<ol id=\"fs-id1167339344897\" type=\"1\">\n<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\n<li>Factor the numerators and denominators.<\/li>\n<li>Multiply the numerators.<\/li>\n<li>Multiply the denominators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_03\" class=\"textbox examples\">\n<div id=\"fs-id1167339296449\">\n<div id=\"fs-id1167339296451\">\n<h3>Dividing Rational Expressions<\/h3>\n<p>Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div id=\"fs-id1167339281593\" class=\"unnumbered aligncenter\">[latex]\\frac{2{x}^{2}+x-6}{{x}^{2}-1}\u00f7\\frac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339185775\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1167339185778\" class=\"unnumbered aligncenter\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}\u00f7\\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339137930\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_03\">\n<div id=\"fs-id1167339432773\">\n<p id=\"fs-id1167339432774\">Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div id=\"fs-id1167339432777\" class=\"unnumbered aligncenter\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x-28}\u00f7\\frac{3{x}^{2}-2x-8}{{x}^{2}+5x-14}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339432756\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339432757\">[latex]1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339429240\" class=\"bc-section section\">\n<h3>Adding and Subtracting Rational Expressions<\/h3>\n<p id=\"fs-id1167339429245\">Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let\u2019s look at an example of fraction addition.<\/p>\n<div id=\"fs-id1167339232046\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\frac{5}{24}+\\frac{1}{40}& =& \\frac{25}{120}+\\frac{3}{120}\\hfill \\\\ & =& \\frac{28}{120}\\hfill \\\\ & =& \\frac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339242302\">We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.<\/p>\n<p id=\"fs-id1167339242307\">The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were[latex]\\,\\left(x+3\\right)\\left(x+4\\right)\\,[\/latex]and[latex]\\,\\left(x+4\\right)\\left(x+5\\right),[\/latex]then the LCD would be[latex]\\,\\left(x+3\\right)\\left(x+4\\right)\\left(x+5\\right).[\/latex]<\/p>\n<p id=\"fs-id1167339212596\">Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of[latex]\\,\\left(x+3\\right)\\left(x+4\\right)\\,[\/latex]by[latex]\\,\\frac{x+5}{x+5}\\,[\/latex]and the expression with a denominator of[latex]\\,\\left(x+4\\right)\\left(x+5\\right)\\,[\/latex]by[latex]\\,\\frac{x+3}{x+3}.[\/latex]<\/p>\n<div id=\"fs-id1167339231542\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339231550\"><strong>Given two rational expressions, add or subtract them.<\/strong><\/p>\n<ol id=\"fs-id1167339231554\" type=\"1\">\n<li>Factor the numerator and denominator.<\/li>\n<li>Find the LCD of the expressions.<\/li>\n<li>Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\n<li>Add or subtract the numerators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_04\" class=\"textbox examples\">\n<div id=\"fs-id1167339212435\">\n<div id=\"fs-id1167339228248\">\n<h3>Adding Rational Expressions<\/h3>\n<p id=\"fs-id1167339228254\">Add the rational expressions:<\/p>\n<div id=\"fs-id1167339228257\" class=\"unnumbered aligncenter\">[latex]\\frac{5}{x}+\\frac{6}{y}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339429820\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339437713\">First, we have to find the LCD. In this case, the LCD will be[latex]\\,xy.\\,[\/latex]We then multiply each expression by the appropriate form of 1 to obtain[latex]\\,xy\\,[\/latex]as the denominator for each fraction.<\/p>\n<div id=\"fs-id1167339212482\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{5}{x}\\cdot \\frac{y}{y}+\\frac{6}{y}\\cdot \\frac{x}{x}\\\\ \\frac{5y}{xy}+\\frac{6x}{xy}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339331048\">Now that the expressions have the same denominator, we simply add the numerators to find the sum.<\/p>\n<div id=\"fs-id1167339331051\" class=\"unnumbered aligncenter\">[latex]\\frac{6x+5y}{xy}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339224193\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339212767\">Multiplying by[latex]\\,\\frac{y}{y}\\,[\/latex]or[latex]\\,\\frac{x}{x}\\,[\/latex]does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_06_05\" class=\"textbox examples\">\n<div id=\"fs-id1167339306374\">\n<div id=\"fs-id1167339306376\">\n<h3>Subtracting Rational Expressions<\/h3>\n<p id=\"fs-id1167339306381\">Subtract the rational expressions:<\/p>\n<div class=\"unnumbered\">[latex]\\frac{6}{{x}^{2}+4x+4}-\\frac{2}{{x}^{2}-4}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339435145\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1167339435147\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}\\frac{6}{{\\left(x+2\\right)}^{2}}-\\frac{2}{\\left(x+2\\right)\\left(x-2\\right)}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Factor}.\\hfill \\\\ \\frac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\frac{x-2}{x-2}-\\frac{2}{\\left(x+2\\right)\\left(x-2\\right)}\\cdot \\frac{x+2}{x+2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply each fraction to get LCD as denominator}.\\hfill \\\\ \\frac{6\\left(x-2\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}-\\frac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply}.\\hfill \\\\ \\frac{6x-12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Apply distributive property}.\\hfill \\\\ \\frac{4x-16}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Subtract}.\\hfill \\\\ \\frac{4\\left(x-4\\right)}{{\\left(x+2\\right)}^{2}\\left(x-2\\right)}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243148\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339243155\"><strong>Do we have to use the LCD to add or subtract rational expressions?<\/strong><\/p>\n<p id=\"fs-id1167339243160\"><em>No. Any common denominator will work, but it is easiest to use the LCD.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1167339243167\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_04\">\n<div id=\"fs-id1167339243178\">\n<p id=\"fs-id1167339243180\">Subtract the rational expressions:[latex]\\,\\frac{3}{x+5}-\\frac{1}{x-3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339432621\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339432622\">[latex]\\frac{2\\left(x-7\\right)}{\\left(x+5\\right)\\left(x-3\\right)}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Simplifying Complex Rational Expressions<\/h3>\n<p>A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression[latex]\\,\\frac{a}{\\frac{1}{b}+c}\\,[\/latex]can be simplified by rewriting the numerator as the fraction[latex]\\,\\frac{a}{1}\\,[\/latex]and combining the expressions in the denominator as[latex]\\,\\frac{1+bc}{b}.\\,[\/latex]We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get[latex]\\,\\frac{a}{1}\\cdot \\frac{b}{1+bc},[\/latex]which is equal to[latex]\\,\\frac{ab}{1+bc}.[\/latex]<\/p>\n<div id=\"fs-id1167339196667\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339196674\"><strong>Given a complex rational expression, simplify it.<\/strong><\/p>\n<ol id=\"fs-id1167339196678\" type=\"1\">\n<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n<li>Rewrite as the numerator divided by the denominator.<\/li>\n<li>Rewrite as multiplication.<\/li>\n<li>Multiply.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1167339196712\">\n<div id=\"fs-id1167339196714\">\n<h3>Simplifying Complex Rational Expressions<\/h3>\n<p id=\"fs-id1167339196719\">Simplify:[latex]\\frac{y+\\frac{1}{x}}{\\frac{x}{y}}[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1167339259637\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339259639\">Begin by combining the expressions in the numerator into one expression.<\/p>\n<div id=\"fs-id1167339259642\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}y\\cdot \\frac{x}{x}+\\frac{1}{x}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Multiply by }\\frac{x}{x}\\text{ to get LCD as denominator}.\\hfill \\\\ \\frac{xy}{x}+\\frac{1}{x}\\hfill & \\\\ \\frac{xy+1}{x}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339306524\">Now the numerator is a single rational expression and the denominator is a single rational expression.<\/p>\n<div id=\"fs-id1167339306528\" class=\"unnumbered aligncenter\">[latex]\\frac{\\frac{xy+1}{x}}{\\frac{x}{y}}[\/latex]<\/div>\n<p id=\"fs-id1167339306577\">We can rewrite this as division, and then multiplication.<\/p>\n<div id=\"fs-id1167339306580\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}\\frac{xy+1}{x}\u00f7\\frac{x}{y}\\hfill & \\\\ \\frac{xy+1}{x}\\cdot \\frac{y}{x}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\frac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223469\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_06_05\">\n<div id=\"fs-id1167339223480\">\n<p id=\"fs-id1167339223481\">Simplify:[latex]\\frac{\\frac{x}{y}-\\frac{y}{x}}{y}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339286463\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339286464\">[latex]\\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339286517\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339286524\"><strong>Can a complex rational expression always be simplified?<\/strong><\/p>\n<p id=\"fs-id1167339286529\"><em>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1167339286537\" class=\"precalculus media\">\n<p id=\"fs-id1167339286544\">Access these online resources for additional instruction and practice with rational expressions.<\/p>\n<ul id=\"fs-id1167339286548\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/simpratexpress\">Simplify Rational Expressions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/multdivratex\">Multiply and Divide Rational Expressions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/addsubratex\">Add and Subtract Rational Expressions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/complexfract\">Simplify a Complex Fraction<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230898\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1167339230905\">\n<li>Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See <a class=\"autogenerated-content\" href=\"#Example_01_06_01\">(Figure)<\/a>.<\/li>\n<li>We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See <a class=\"autogenerated-content\" href=\"#Example_01_06_02\">(Figure)<\/a>.<\/li>\n<li>To divide rational expressions, multiply by the reciprocal of the second expression. See <a class=\"autogenerated-content\" href=\"#Example_01_06_03\">(Figure)<\/a>.<\/li>\n<li>Adding or subtracting rational expressions requires finding a common denominator. See <a class=\"autogenerated-content\" href=\"#Example_01_06_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_01_06_05\">(Figure)<\/a>.<\/li>\n<li>Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See <a class=\"autogenerated-content\" href=\"#Example_01_06_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167339230956\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1167339230960\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1167339230966\">\n<div id=\"fs-id1167339230967\">\n<p id=\"fs-id1167339230968\">How can you use factoring to simplify rational expressions?<\/p>\n<\/div>\n<div id=\"fs-id1167339230971\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339230972\">You can factor the numerator and denominator to see if any of the terms can cancel one another out.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339230978\">How do you use the LCD to combine two rational expressions?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230982\">\n<div id=\"fs-id1167339230983\">\n<p id=\"fs-id1167339230984\">Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.<\/p>\n<\/div>\n<div id=\"fs-id1167339230988\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339230989\">True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230995\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1167339231001\">For the following exercises, simplify the rational expressions.<\/p>\n<div id=\"fs-id1167339231004\">\n<div id=\"fs-id1167339231005\">\n<p id=\"fs-id1167339231006\">[latex]\\frac{{x}^{2}-16}{{x}^{2}-5x+4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339199459\">\n<div id=\"fs-id1167339199460\">\n<p id=\"fs-id1167339199461\">[latex]\\frac{{y}^{2}+10y+25}{{y}^{2}+11y+30}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339199522\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339199523\">[latex]\\frac{y+5}{y+6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339199556\">\n<div id=\"fs-id1167339199557\">\n<p id=\"fs-id1167339199558\">[latex]\\frac{6{a}^{2}-24a+24}{6{a}^{2}-24}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220266\">\n<div id=\"fs-id1167339220267\">\n<p id=\"fs-id1167339220268\">[latex]\\frac{9{b}^{2}+18b+9}{3b+3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339220319\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339220320\">[latex]3b+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220338\">\n<div id=\"fs-id1167339220339\">\n<p id=\"fs-id1167339220340\">[latex]\\frac{m-12}{{m}^{2}-144}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339428487\">\n<div id=\"fs-id1167339428488\">\n<p id=\"fs-id1167339428489\">[latex]\\frac{2{x}^{2}+7x-4}{4{x}^{2}+2x-2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339428554\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339428556\">[latex]\\frac{x+4}{2x+2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339428591\">\n<div id=\"fs-id1167339428592\">\n<p id=\"fs-id1167339428593\">[latex]\\frac{6{x}^{2}+5x-4}{3{x}^{2}+19x+20}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339226352\">\n<div id=\"fs-id1167339226353\">\n<p id=\"fs-id1167339226354\">[latex]\\frac{{a}^{2}+9a+18}{{a}^{2}+3a-18}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339226415\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339226416\">[latex]\\frac{a+3}{a-3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196454\">\n<div id=\"fs-id1167339196456\">\n<p id=\"fs-id1167339196457\">[latex]\\frac{3{c}^{2}+25c-18}{3{c}^{2}-23c+14}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339196524\">[latex]\\frac{12{n}^{2}-29n-8}{28{n}^{2}-5n-3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339260411\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339260412\">[latex]\\frac{3n-8}{7n-3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339260450\">For the following exercises, multiply the rational expressions and express the product in simplest form.<\/p>\n<div id=\"fs-id1167339260454\">\n<div id=\"fs-id1167339260455\">\n<p id=\"fs-id1167339260456\">[latex]\\frac{{x}^{2}-x-6}{2{x}^{2}+x-6}\\cdot \\frac{2{x}^{2}+7x-15}{{x}^{2}-9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339303584\">\n<div id=\"fs-id1167339303585\">\n<p id=\"fs-id1167339303586\">[latex]\\frac{{c}^{2}+2c-24}{{c}^{2}+12c+36}\\cdot \\frac{{c}^{2}-10c+24}{{c}^{2}-8c+16}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339303701\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339303702\">[latex]\\frac{c-6}{c+6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339218644\">[latex]\\frac{2{d}^{2}+9d-35}{{d}^{2}+10d+21}\\cdot \\frac{3{d}^{2}+2d-21}{3{d}^{2}+14d-49}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339218766\">\n<div id=\"fs-id1167339218767\">\n<p id=\"fs-id1167339218768\">[latex]\\frac{10{h}^{2}-9h-9}{2{h}^{2}-19h+24}\\cdot \\frac{{h}^{2}-16h+64}{5{h}^{2}-37h-24}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339239122\">[latex]1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239131\">\n<div id=\"fs-id1167339239132\">\n<p id=\"fs-id1167339239133\">[latex]\\frac{6{b}^{2}+13b+6}{4{b}^{2}-9}\\cdot \\frac{6{b}^{2}+31b-30}{18{b}^{2}-3b-10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339321412\">\n<div id=\"fs-id1167339321413\">\n<p id=\"fs-id1167339321414\">[latex]\\frac{2{d}^{2}+15d+25}{4{d}^{2}-25}\\cdot \\frac{2{d}^{2}-15d+25}{25{d}^{2}-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339344433\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339344434\">[latex]\\frac{{d}^{2}-25}{25{d}^{2}-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339344484\">\n<div id=\"fs-id1167339344485\">\n<p id=\"fs-id1167339344486\">[latex]\\frac{6{x}^{2}-5x-50}{15{x}^{2}-44x-20}\\cdot \\frac{20{x}^{2}-7x-6}{2{x}^{2}+9x+10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138825\">\n<div id=\"fs-id1167339138826\">\n<p id=\"fs-id1167339138827\">[latex]\\frac{{t}^{2}-1}{{t}^{2}+4t+3}\\cdot \\frac{{t}^{2}+2t-15}{{t}^{2}-4t+3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339138936\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339138937\">[latex]\\frac{t+5}{t+3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273802\">\n<div id=\"fs-id1167339273803\">\n<p id=\"fs-id1167339273804\">[latex]\\frac{2{n}^{2}-n-15}{6{n}^{2}+13n-5}\\cdot \\frac{12{n}^{2}-13n+3}{4{n}^{2}-15n+9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273926\">\n<div id=\"fs-id1167339273927\">\n<p id=\"fs-id1167339273928\">[latex]\\frac{36{x}^{2}-25}{6{x}^{2}+65x+50}\\cdot \\frac{3{x}^{2}+32x+20}{18{x}^{2}+27x+10}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339428060\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339428061\">[latex]\\frac{6x-5}{6x+5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339428098\">For the following exercises, divide the rational expressions.<\/p>\n<div id=\"fs-id1167339428102\">\n<div id=\"fs-id1167339428103\">\n<p id=\"fs-id1167339428104\">[latex]\\frac{3{y}^{2}-7y-6}{2{y}^{2}-3y-9}\u00f7\\frac{{y}^{2}+y-2}{2{y}^{2}+y-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339227755\">\n<div id=\"fs-id1167339227756\">\n<p id=\"fs-id1167339227757\">[latex]\\frac{6{p}^{2}+p-12}{8{p}^{2}+18p+9}\u00f7\\frac{6{p}^{2}-11p+4}{2{p}^{2}+11p-6}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339314058\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339314059\">[latex]\\frac{p+6}{4p+3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339314094\">\n<div id=\"fs-id1167339314096\">\n<p id=\"fs-id1167339314097\">[latex]\\frac{{q}^{2}-9}{{q}^{2}+6q+9}\u00f7\\frac{{q}^{2}-2q-3}{{q}^{2}+2q-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339314205\">\n<div id=\"fs-id1167339314206\">\n<p id=\"fs-id1167339314208\">[latex]\\frac{18{d}^{2}+77d-18}{27{d}^{2}-15d+2}\u00f7\\frac{3{d}^{2}+29d-44}{9{d}^{2}-15d+4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339214028\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339214029\">[latex]\\frac{2d+9}{d+11}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214064\">\n<div id=\"fs-id1167339214065\">\n<p id=\"fs-id1167339214066\">[latex]\\frac{16{x}^{2}+18x-55}{32{x}^{2}-36x-11}\u00f7\\frac{2{x}^{2}+17x+30}{4{x}^{2}+25x+6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339222916\">\n<div id=\"fs-id1167339222917\">\n<p id=\"fs-id1167339222918\">[latex]\\frac{144{b}^{2}-25}{72{b}^{2}-6b-10}\u00f7\\frac{18{b}^{2}-21b+5}{36{b}^{2}-18b-10}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"fs-id1167339222916\">\n<div>\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\frac{12b+5}{3b-1}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339280911\">\n<div id=\"fs-id1167339280912\">\n<p id=\"fs-id1167339280914\">[latex]\\frac{16{a}^{2}-24a+9}{4{a}^{2}+17a-15}\u00f7\\frac{16{a}^{2}-9}{4{a}^{2}+11a+6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339281031\">\n<div>\n<p id=\"fs-id1167339281033\">[latex]\\frac{22{y}^{2}+59y+10}{12{y}^{2}+28y-5}\u00f7\\frac{11{y}^{2}+46y+8}{24{y}^{2}-10y+1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339225605\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339225606\">[latex]\\frac{4y-1}{y+4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339225642\">\n<div id=\"fs-id1167339225643\">\n<p id=\"fs-id1167339225644\">[latex]\\frac{9{x}^{2}+3x-20}{3{x}^{2}-7x+4}\u00f7\\frac{6{x}^{2}+4x-10}{{x}^{2}-2x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339225767\">For the following exercises, add and subtract the rational expressions, and then simplify.<\/p>\n<div id=\"fs-id1167339225771\">\n<div id=\"fs-id1167339225772\">\n<p id=\"fs-id1167339225774\">[latex]\\frac{4}{x}+\\frac{10}{y}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339317636\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339317638\">[latex]\\frac{10x+4y}{xy}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317673\">\n<div>\n<p id=\"fs-id1167339317675\">[latex]\\frac{12}{2q}-\\frac{6}{3p}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317719\">\n<div id=\"fs-id1167339317720\">\n<p id=\"fs-id1167339317721\">[latex]\\frac{4}{a+1}+\\frac{5}{a-3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339317766\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339317767\">[latex]\\frac{9a-7}{{a}^{2}-2a-3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317816\">\n<div id=\"fs-id1167339317817\">\n<p id=\"fs-id1167339317818\">[latex]\\frac{c+2}{3}-\\frac{c-4}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339317863\">\n<div id=\"fs-id1167339317864\">\n<p id=\"fs-id1167339317865\">[latex]\\frac{y+3}{y-2}+\\frac{y-3}{y+1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339315713\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339315714\">[latex]\\frac{2{y}^{2}-y+9}{{y}^{2}-y-2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339315772\">\n<div id=\"fs-id1167339315773\">\n<p id=\"fs-id1167339315774\">[latex]\\frac{x-1}{x+1}-\\frac{2x+3}{2x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339315838\">\n<div id=\"fs-id1167339315840\">\n<p id=\"fs-id1167339315841\">[latex]\\frac{3z}{z+1}+\\frac{2z+5}{z-2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339315900\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339315901\">[latex]\\frac{5{z}^{2}+z+5}{{z}^{2}-z-2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239217\">\n<div>[latex]\\frac{4p}{p+1}-\\frac{p+1}{4p}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339239275\">\n<div id=\"fs-id1167339239276\">\n<p id=\"fs-id1167339239277\">[latex]\\frac{x}{x+1}+\\frac{y}{y+1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339239322\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339239323\">[latex]\\frac{x+2xy+y}{x+xy+y+1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339239375\">For the following exercises, simplify the rational expression.<\/p>\n<div id=\"fs-id1167339239378\">\n<div id=\"fs-id1167339239380\">\n<p id=\"fs-id1167339239381\">[latex]\\frac{\\frac{6}{y}-\\frac{4}{x}}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239420\">\n<div id=\"fs-id1167339239421\">\n<p id=\"fs-id1167339239422\">[latex]\\frac{\\frac{2}{a}+\\frac{7}{b}}{b}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339437819\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\frac{2b+7a}{a{b}^{2}}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339437862\">\n<div id=\"fs-id1167339437863\">\n<p id=\"fs-id1167339437864\">[latex]\\frac{\\frac{x}{4}-\\frac{p}{8}}{p}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339437904\">\n<div id=\"fs-id1167339437905\">\n<p id=\"fs-id1167339437906\">[latex]\\frac{\\frac{3}{a}+\\frac{b}{6}}{\\frac{2b}{3a}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339437965\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339437966\">[latex]\\frac{18+ab}{4b}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339438000\">\n<div id=\"fs-id1167339438001\">\n<p id=\"fs-id1167339438002\">[latex]\\frac{\\frac{3}{x+1}+\\frac{2}{x-1}}{\\frac{x-1}{x+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339438081\">\n<div id=\"fs-id1167339240839\">\n<p id=\"fs-id1167339240840\">[latex]\\frac{\\frac{a}{b}-\\frac{b}{a}}{\\frac{a+b}{ab}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339240902\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339240903\">[latex]a-b[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339240919\">\n<div id=\"fs-id1167339240920\">\n<p id=\"fs-id1167339240921\">[latex]\\frac{\\frac{2x}{3}+\\frac{4x}{7}}{\\frac{x}{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339240980\">\n<div id=\"fs-id1167339240982\">\n<p id=\"fs-id1167339240983\">[latex]\\frac{\\frac{2c}{c+2}+\\frac{c-1}{c+1}}{\\frac{2c+1}{c+1}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339241076\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339241078\">[latex]\\frac{3{c}^{2}+3c-2}{2{c}^{2}+5c+2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1167339433020\">[latex]\\frac{\\frac{x}{y}-\\frac{y}{x}}{\\frac{x}{y}+\\frac{y}{x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433082\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1167339433087\">\n<div id=\"fs-id1167339433088\">\n<p id=\"fs-id1167339433089\">Brenda is placing tile on her bathroom floor. The area of the floor is[latex]\\,15{x}^{2}-8x-7\\,[\/latex]ft<sup>2<\/sup>. The area of one tile is[latex]\\,{x}^{2}-2x+1{\\text{ft}}^{2}.\\,[\/latex]To find the number of tiles needed, simplify the rational expression:[latex]\\,\\frac{15{x}^{2}-8x-7}{{x}^{2}-2x+1}.[\/latex]<\/p>\n<p><span id=\"fs-id1167339433096\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132210\/CNX_CAT_Figure_01_06_201.jpg\" alt=\"A rectangle that\u2019s labeled: Area = fifteen times x squared minus eight times x minus seven.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1167339433108\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339433109\">[latex]\\frac{15x+7}{x-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433146\">\n<div id=\"fs-id1167339433147\">\n<p id=\"fs-id1167339433148\">The area of Sandy\u2019s yard is[latex]\\,25{x}^{2}-625\\,[\/latex]ft<sup>2<\/sup>. A patch of sod has an area of[latex]\\,{x}^{2}-10x+25\\,[\/latex]ft<sup>2<\/sup>. Divide the two areas and simplify to find how many pieces of sod Sandy needs to cover her yard.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433218\">\n<div id=\"fs-id1167339433219\">\n<p id=\"fs-id1167339433220\">Aaron wants to mulch his garden. His garden is[latex]\\,{x}^{2}+18x+81\\,[\/latex]ft<sup>2<\/sup>. One bag of mulch covers[latex]\\,{x}^{2}-81\\,[\/latex]ft<sup>2<\/sup>. Divide the expressions and simplify to find how many bags of mulch Aaron needs to mulch his garden.<\/p>\n<\/div>\n<div id=\"fs-id1167339433288\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339433289\">[latex]\\frac{x+9}{x-9}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299394\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1167339299400\">For the following exercises, perform the given operations and simplify.<\/p>\n<div id=\"fs-id1167339299403\">\n<div id=\"fs-id1167339299404\">\n<p id=\"fs-id1167339299405\">[latex]\\frac{{x}^{2}+x-6}{{x}^{2}-2x-3}\\cdot \\frac{2{x}^{2}-3x-9}{{x}^{2}-x-2}\u00f7\\frac{10{x}^{2}+27x+18}{{x}^{2}+2x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299574\">\n<div id=\"fs-id1167339299576\">\n<p id=\"fs-id1167339299577\">[latex]\\frac{\\frac{3{y}^{2}-10y+3}{3{y}^{2}+5y-2}\\cdot \\frac{2{y}^{2}-3y-20}{2{y}^{2}-y-15}}{y-4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339299716\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339299717\">[latex]\\frac{1}{y+2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219221\">\n<div id=\"fs-id1167339219222\">\n<p id=\"fs-id1167339219223\">[latex]\\frac{\\frac{4a+1}{2a-3}+\\frac{2a-3}{2a+3}}{\\frac{4{a}^{2}+9}{a}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219330\">\n<div id=\"fs-id1167339219331\">\n<p id=\"fs-id1167339219332\">[latex]\\frac{{x}^{2}+7x+12}{{x}^{2}+x-6}\u00f7\\frac{3{x}^{2}+19x+28}{8{x}^{2}-4x-24}\u00f7\\frac{2{x}^{2}+x-3}{3{x}^{2}+4x-7}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339219508\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339219509\">[latex]4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219520\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1167339219527\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/d0a86917-6447-4b49-bed1-efb5c352cc77\">Real Numbers: Algebra Essentials<\/a><\/h4>\n<p id=\"fs-id1167339219537\">For the following exercises, perform the given operations.<\/p>\n<div id=\"fs-id1167339219540\">\n<div id=\"fs-id1167339331066\">\n<p id=\"fs-id1167339331067\">[latex]{\\left(5-3\\cdot 2\\right)}^{2}-6[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339331113\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339331114\">[latex]-5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331128\">\n<div id=\"fs-id1167339331129\">\n<p id=\"fs-id1167339331130\">[latex]64\u00f7\\left(2\\cdot 8\\right)+14\u00f77[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331169\">\n<div id=\"fs-id1167339331170\">\n<p id=\"fs-id1167339331171\">[latex]2\\cdot {5}^{2}+6\u00f72[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339331203\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339331204\">53<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339331208\">For the following exercises, solve the equation.<\/p>\n<div id=\"fs-id1167339331211\">\n<div id=\"fs-id1167339331212\">\n<p id=\"fs-id1167339331213\">[latex]5x+9=-11[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331237\">\n<div id=\"fs-id1167339331238\">\n<p id=\"fs-id1167339331240\">[latex]2y+{4}^{2}=64[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339331269\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339331270\">[latex]y=24[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339331286\">For the following exercises, simplify the expression.<\/p>\n<div id=\"fs-id1167339331290\">\n<div id=\"fs-id1167339331291\">\n<p id=\"fs-id1167339331292\">[latex]9\\left(y+2\\right)\u00f73\\cdot 2+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331333\">\n<div id=\"fs-id1167339331334\">\n<p id=\"fs-id1167339331336\">[latex]3m\\left(4+7\\right)-m[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339331371\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339331372\">[latex]32m[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339331386\">For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.<\/p>\n<div id=\"fs-id1167339331390\">\n<div id=\"fs-id1167339331391\">\n<p id=\"fs-id1167339331392\">11<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331395\">\n<div id=\"fs-id1167339331396\">\n<p id=\"fs-id1167339331398\">0<\/p>\n<\/div>\n<div id=\"fs-id1167339331401\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339331402\">whole<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339331405\">\n<div id=\"fs-id1167339331406\">\n<p id=\"fs-id1167339331407\">[latex]\\frac{5}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432116\">\n<div id=\"fs-id1167339432117\">\n<p id=\"fs-id1167339432118\">[latex]\\sqrt{11}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339432136\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339432137\">irrational<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432142\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/065c3182-adac-48fc-a3ab-03d487982dd5\">Exponents and Scientific Notation<\/a><\/h4>\n<p id=\"fs-id1167339432152\">For the following exercises, simplify the expression.<\/p>\n<div id=\"fs-id1167339432155\">\n<div id=\"fs-id1167339432156\">\n<p id=\"fs-id1167339432157\">[latex]{2}^{2}\\cdot {2}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432187\">\n<div id=\"fs-id1167339432188\">\n<p id=\"fs-id1167339432189\">[latex]\\frac{{4}^{5}}{{4}^{3}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339432227\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339432228\">[latex]16[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432240\">\n<div id=\"fs-id1167339432241\">\n<p id=\"fs-id1167339432242\">[latex]{\\left(\\frac{{a}^{2}}{{b}^{3}}\\right)}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432299\">\n<div id=\"fs-id1167339432300\">\n<p id=\"fs-id1167339432301\">[latex]\\frac{6{a}^{2}\\cdot {a}^{0}}{2{a}^{-4}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339432360\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339432361\">[latex]{a}^{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432379\">\n<div id=\"fs-id1167339432380\">\n<p id=\"fs-id1167339432381\">[latex]\\frac{{\\left(xy\\right)}^{4}}{{y}^{3}}\\cdot \\frac{2}{{x}^{5}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339432456\">\n<div id=\"fs-id1167339432457\">\n<p id=\"fs-id1167339432458\">[latex]\\frac{{4}^{-2}{x}^{3}{y}^{-3}}{2{x}^{0}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339295802\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339295803\">[latex]\\frac{{x}^{3}}{32{y}^{3}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339295842\">\n<div id=\"fs-id1167339295843\">\n<p id=\"fs-id1167339295844\">[latex]{\\left(\\frac{2{x}^{2}}{y}\\right)}^{-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339295900\">\n<div id=\"fs-id1167339295901\">\n<p id=\"fs-id1167339295902\">[latex]\\left(\\frac{16{a}^{3}}{{b}^{2}}\\right){\\left(4a{b}^{-1}\\right)}^{-2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339295996\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339295997\">[latex]a[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296005\">\n<div id=\"fs-id1167339296006\">\n<p id=\"fs-id1167339296007\">Write the number in standard notation:[latex]\\,2.1314\\,\u00d7\\,{10}^{-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296042\">\n<div id=\"fs-id1167339296043\">\n<p id=\"fs-id1167339296044\">Write the number in scientific notation: 16,340,000<\/p>\n<\/div>\n<div id=\"fs-id1167339296047\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339296048\">[latex]1.634\\,\u00d7\\,{10}^{7}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296075\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/1834bc64-7094-4df1-a530-b3166a295697\">Radicals and Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167339296085\">For the following exercises, find the principal square root.<\/p>\n<div id=\"fs-id1167339296088\">\n<div id=\"fs-id1167339296089\">\n<p id=\"fs-id1167339296090\">[latex]\\sqrt{121}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296108\">\n<div id=\"fs-id1167339296110\">\n<p id=\"fs-id1167339296111\">[latex]\\sqrt{196}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339296129\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339296130\">14<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296133\">\n<div id=\"fs-id1167339296134\">\n<p id=\"fs-id1167339296135\">[latex]\\sqrt{361}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296153\">\n<div id=\"fs-id1167339296154\">\n<p id=\"fs-id1167339296155\">[latex]\\sqrt{75}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339296174\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339296175\">[latex]5\\sqrt{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296192\">\n<div id=\"fs-id1167339296193\">\n<p id=\"fs-id1167339296194\">[latex]\\sqrt{162}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339296212\">\n<div id=\"fs-id1167339296213\">\n<p id=\"fs-id1167339296214\">[latex]\\sqrt{\\frac{32}{25}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339296245\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339296246\">[latex]\\frac{4\\sqrt{2}}{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206368\">\n<div id=\"fs-id1167339206369\">\n<p id=\"fs-id1167339206370\">[latex]\\sqrt{\\frac{80}{81}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206401\">\n<div id=\"fs-id1167339206402\">\n<p id=\"fs-id1167339206403\">[latex]\\sqrt{\\frac{49}{1250}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339206434\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339206435\">[latex]\\frac{7\\sqrt{2}}{50}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206465\">\n<div id=\"fs-id1167339206466\">\n<p id=\"fs-id1167339206467\">[latex]\\frac{2}{4+\\sqrt{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206496\">\n<div id=\"fs-id1167339206497\">\n<p id=\"fs-id1167339206498\">[latex]4\\sqrt{3}+6\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339206525\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339206526\">[latex]10\\sqrt{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206544\">\n<div id=\"fs-id1167339206545\">\n<p id=\"fs-id1167339206546\">[latex]12\\sqrt{5}-13\\sqrt{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206573\">\n<div id=\"fs-id1167339206574\">\n<p id=\"fs-id1167339206576\">[latex]\\sqrt[5]{-243}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339206599\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339206600\">[latex]-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206614\">\n<div id=\"fs-id1167339206615\">\n<p id=\"fs-id1167339206616\">[latex]\\frac{\\sqrt[3]{250}}{\\sqrt[3]{-8}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206663\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/f7978ad8-ed27-4fe4-8a19-26a031ba97ad\">Polynomials<\/a><\/h4>\n<p id=\"fs-id1167339206673\">For the following exercises, perform the given operations and simplify.<\/p>\n<div id=\"fs-id1167339206677\">\n<div id=\"fs-id1167339206678\">\n<p id=\"fs-id1167339206679\">[latex]\\left(3{x}^{3}+2x-1\\right)+\\left(4{x}^{2}-2x+7\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339206757\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339206758\">[latex]3{x}^{3}+4{x}^{2}+6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206797\">\n<div id=\"fs-id1167339206798\">\n<p id=\"fs-id1167339206799\">[latex]\\left(2y+1\\right)-\\left(2{y}^{2}-2y-5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339206863\">\n<div id=\"fs-id1167339206864\">\n<p id=\"fs-id1167339206865\">[latex]\\left(2{x}^{2}+3x-6\\right)+\\left(3{x}^{2}-4x+9\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339297739\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339297740\">[latex]5{x}^{2}-x+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297770\">\n<div id=\"fs-id1167339297771\">\n<p id=\"fs-id1167339297772\">[latex]\\left(6{a}^{2}+3a+10\\right)-\\left(6{a}^{2}-3a+5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297850\">\n<div id=\"fs-id1167339297851\">\n<p id=\"fs-id1167339297852\">[latex]\\left(k+3\\right)\\left(k-6\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339297896\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339297897\">[latex]{k}^{2}-3k-18[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297927\">\n<div id=\"fs-id1167339297928\">\n<p id=\"fs-id1167339297929\">[latex]\\left(2h+1\\right)\\left(3h-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339297977\">\n<div id=\"fs-id1167339297978\">\n<p id=\"fs-id1167339297979\">[latex]\\left(x+1\\right)\\left({x}^{2}+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339298030\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339298032\">[latex]{x}^{3}+{x}^{2}+x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298071\">\n<div id=\"fs-id1167339298072\">\n<p id=\"fs-id1167339298073\">[latex]\\left(m-2\\right)\\left({m}^{2}+2m-3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298131\">\n<div id=\"fs-id1167339298132\">\n<p id=\"fs-id1167339298133\">[latex]\\left(a+2b\\right)\\left(3a-b\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339298181\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339298182\">[latex]3{a}^{2}+5ab-2{b}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298225\">\n<div id=\"fs-id1167339298226\">\n<p id=\"fs-id1167339298227\">[latex]\\left(x+y\\right)\\left(x-y\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339298272\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/49cf2d69-1d37-49aa-9e61-16da4c52ce37\">Factoring Polynomials<\/a><\/h4>\n<p id=\"fs-id1167339298282\">For the following exercises, find the greatest common factor.<\/p>\n<div id=\"fs-id1167339298285\">\n<div id=\"fs-id1167339298286\">\n<p id=\"fs-id1167339298287\">[latex]81p+9pq-27{p}^{2}{q}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339298332\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339298333\">[latex]9p[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308685\">\n<div id=\"fs-id1167339308686\">\n<p id=\"fs-id1167339308687\">[latex]12{x}^{2}y+4x{y}^{2}-18xy[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308735\">\n<div id=\"fs-id1167339308736\">\n<p id=\"fs-id1167339308737\">[latex]88{a}^{3}b+4{a}^{2}b-144{a}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339308789\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339308790\">[latex]4{a}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339308811\">For the following exercises, factor the polynomial.<\/p>\n<div id=\"fs-id1167339308814\">\n<div id=\"fs-id1167339308815\">\n<p id=\"fs-id1167339308816\">[latex]2{x}^{2}-9x-18[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308848\">\n<div id=\"fs-id1167339308849\">\n<p id=\"fs-id1167339308850\">[latex]8{a}^{2}+30a-27[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339308882\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339308883\">[latex]\\left(4a-3\\right)\\left(2a+9\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308931\">\n<div id=\"fs-id1167339308932\">\n<p id=\"fs-id1167339308933\">[latex]{d}^{2}-5d-66[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339308963\">\n<div id=\"fs-id1167339308964\">\n<p id=\"fs-id1167339308965\">[latex]{x}^{2}+10x+25[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339308995\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339308996\">[latex]{\\left(x+5\\right)}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309032\">\n<div id=\"fs-id1167339309034\">\n<p id=\"fs-id1167339309035\">[latex]{y}^{2}-6y+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309064\">\n<div id=\"fs-id1167339309066\">\n<p id=\"fs-id1167339309067\">[latex]4{h}^{2}-12hk+9{k}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339309109\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339309110\">[latex]{\\left(2h-3k\\right)}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309151\">\n<div id=\"fs-id1167339309152\">\n<p id=\"fs-id1167339309153\">[latex]361{x}^{2}-121[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309179\">\n<div id=\"fs-id1167339309180\">\n<p id=\"fs-id1167339309181\">[latex]{p}^{3}+216[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339309204\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339309205\">[latex]\\left(p+6\\right)\\left({p}^{2}-6p+36\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309263\">\n<div id=\"fs-id1167339309264\">\n<p id=\"fs-id1167339309265\">[latex]8{x}^{3}-125[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309291\">\n<div id=\"fs-id1167339309292\">\n<p id=\"fs-id1167339309293\">[latex]64{q}^{3}-27{p}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339309327\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339309328\">[latex]\\left(4q-3p\\right)\\left(16{q}^{2}+12pq+9{p}^{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309403\">\n<div id=\"fs-id1167339309404\">\n<p id=\"fs-id1167339309405\">[latex]4x{\\left(x-1\\right)}^{-\\frac{1}{4}}+3{\\left(x-1\\right)}^{\\frac{3}{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339309499\">\n<div id=\"fs-id1167339309500\">\n<p id=\"fs-id1167339309501\">[latex]3p{\\left(p+3\\right)}^{\\frac{1}{3}}-8{\\left(p+3\\right)}^{\\frac{4}{3}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339433736\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339433737\">[latex]{\\left(p+3\\right)}^{\\frac{1}{3}}\\left(-5p-24\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433805\">\n<div id=\"fs-id1167339433806\">\n<p id=\"fs-id1167339433808\">[latex]4r{\\left(2r-1\\right)}^{-\\frac{2}{3}}-5{\\left(2r-1\\right)}^{\\frac{1}{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339433907\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"\/contents\/45953190-e727-41e0-9e6a-834e092ee148\">Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167339433916\">For the following exercises, simplify the expression.<\/p>\n<div id=\"fs-id1167339433919\">\n<div id=\"fs-id1167339433920\">\n<p id=\"fs-id1167339433922\">[latex]\\frac{{x}^{2}-x-12}{{x}^{2}-8x+16}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339433980\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339433981\">[latex]\\frac{x+3}{x-4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434014\">\n<div id=\"fs-id1167339434015\">\n<p id=\"fs-id1167339434016\">[latex]\\frac{4{y}^{2}-25}{4{y}^{2}-20y+25}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434075\">\n<div id=\"fs-id1167339434076\">\n<p id=\"fs-id1167339434077\">[latex]\\frac{2{a}^{2}-a-3}{2{a}^{2}-6a-8}\\cdot \\frac{5{a}^{2}-19a-4}{10{a}^{2}-13a-3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339434201\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339434202\">[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434220\">\n<div id=\"fs-id1167339434221\">\n<p id=\"fs-id1167339434222\">[latex]\\frac{d-4}{{d}^{2}-9}\\cdot \\frac{d-3}{{d}^{2}-16}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434297\">\n<div id=\"fs-id1167339434298\">\n<p id=\"fs-id1167339434299\">[latex]\\frac{{m}^{2}+5m+6}{2{m}^{2}-5m-3}\u00f7\\frac{2{m}^{2}+3m-9}{4{m}^{2}-4m-3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339434421\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339434422\">[latex]\\frac{m+2}{m-3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434455\">\n<div id=\"fs-id1167339434456\">\n<p id=\"fs-id1167339434457\">[latex]\\frac{4{d}^{2}-7d-2}{6{d}^{2}-17d+10}\u00f7\\frac{8{d}^{2}+6d+1}{6{d}^{2}+7d-10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434581\">\n<div id=\"fs-id1167339434582\">\n<p id=\"fs-id1167339434583\">[latex]\\frac{10}{x}+\\frac{6}{y}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339434616\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339434617\">[latex]\\frac{6x+10y}{xy}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434652\">\n<div id=\"fs-id1167339434653\">\n<p id=\"fs-id1167339434654\">[latex]\\frac{12}{{a}^{2}+2a+1}-\\frac{3}{{a}^{2}-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434724\">\n<div id=\"fs-id1167339434725\">\n<p id=\"fs-id1167339434726\">[latex]\\frac{\\frac{1}{d}+\\frac{2}{c}}{\\frac{6c+12d}{dc}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339434792\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339434793\">[latex]\\frac{1}{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434811\">\n<div id=\"fs-id1167339434812\">\n<p id=\"fs-id1167339434813\">[latex]\\frac{\\frac{3}{x}-\\frac{7}{y}}{\\frac{2}{x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434864\" class=\"practice-test\">\n<h3>Chapter Practice Test<\/h3>\n<p id=\"fs-id1167339434872\">For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.<\/p>\n<div id=\"fs-id1167339434876\">\n<div id=\"fs-id1167339434877\">\n<p id=\"fs-id1167339434878\">[latex]-13[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339434892\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339434893\">rational<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434896\">\n<div id=\"fs-id1167339434897\">\n<p id=\"fs-id1167339434898\">[latex]\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339434913\">For the following exercises, evaluate the equations.<\/p>\n<div id=\"fs-id1167339434917\">\n<div id=\"fs-id1167339434918\">\n<p id=\"fs-id1167339434919\">[latex]2\\left(x+3\\right)-12=18[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339434956\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339434957\">[latex]x=12[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339434973\">\n<div id=\"fs-id1167339434974\">\n<p id=\"fs-id1167339434975\">[latex]y{\\left(3+3\\right)}^{2}-26=10[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1167339435024\">\n<p id=\"fs-id1167339435025\">Write the number in standard notation:[latex]3.1415\\,\u00d7\\,{10}^{6}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339435052\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339435053\">3,141,500<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342169\">\n<div id=\"fs-id1167339342170\">\n<p id=\"fs-id1167339342171\">Write the number in scientific notation: 0.0000000212.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339342174\">For the following exercises, simplify the expression.<\/p>\n<div id=\"fs-id1167339342178\">\n<div id=\"fs-id1167339342179\">\n<p id=\"fs-id1167339342180\">[latex]-2\\cdot {\\left(2+3\\cdot 2\\right)}^{2}+144[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342232\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342233\">[latex]16[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342245\">\n<div id=\"fs-id1167339342246\">\n<p id=\"fs-id1167339342247\">[latex]4\\left(x+3\\right)-\\left(6x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342297\">\n<div id=\"fs-id1167339342298\">\n<p id=\"fs-id1167339342299\">[latex]{3}^{5}\\cdot {3}^{-3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342334\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342336\">9<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342339\">\n<div id=\"fs-id1167339342340\">\n<p id=\"fs-id1167339342341\">[latex]{\\left(\\frac{2}{3}\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342381\">\n<div id=\"fs-id1167339342382\">\n<p id=\"fs-id1167339342383\">[latex]\\frac{8{x}^{3}}{{\\left(2x\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342441\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342442\">[latex]2x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342457\">\n<div id=\"fs-id1167339342458\">\n<p id=\"fs-id1167339342459\">[latex]\\left(16{y}^{0}\\right)2{y}^{-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342506\">\n<div id=\"fs-id1167339342507\">\n<p id=\"fs-id1167339342508\">[latex]\\sqrt{441}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342526\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342527\">21<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342530\">\n<div id=\"fs-id1167339342532\">\n<p id=\"fs-id1167339342533\">[latex]\\sqrt{490}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342551\">\n<div id=\"fs-id1167339342552\">\n<p id=\"fs-id1167339342553\">[latex]\\sqrt{\\frac{9x}{16}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342586\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342587\">[latex]\\frac{3\\sqrt{x}}{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342614\">\n<div id=\"fs-id1167339342615\">\n<p id=\"fs-id1167339342616\">[latex]\\frac{\\sqrt{121{b}^{2}}}{1+\\sqrt{b}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342663\">\n<div id=\"fs-id1167339342664\">\n<p id=\"fs-id1167339342665\">[latex]6\\sqrt{24}+7\\sqrt{54}-12\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342710\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342711\">[latex]21\\sqrt{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342728\">\n<div id=\"fs-id1167339342729\">\n<p id=\"fs-id1167339342730\">[latex]\\frac{\\sqrt[3]{-8}}{\\sqrt[4]{625}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342776\">\n<div id=\"fs-id1167339342777\">\n<p id=\"fs-id1167339342778\">[latex]\\left(13{q}^{3}+2{q}^{2}-3\\right)-\\left(6{q}^{2}+5q-3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339342863\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339342864\">[latex]13{q}^{3}-4{q}^{2}-5q[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342906\">\n<div id=\"fs-id1167339342907\">\n<p id=\"fs-id1167339342908\">[latex]\\left(6{p}^{2}+2p+1\\right)+\\left(9{p}^{2}-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339342980\">\n<div id=\"fs-id1167339342981\">\n<p id=\"fs-id1167339342982\">[latex]\\left(n-2\\right)\\left({n}^{2}-4n+4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339343039\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339343040\">[latex]{n}^{3}-6{n}^{2}+12n-8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343084\">\n<div id=\"fs-id1167339343085\">\n<p id=\"fs-id1167339343086\">[latex]\\left(a-2b\\right)\\left(2a+b\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339343134\">For the following exercises, factor the polynomial.<\/p>\n<div id=\"fs-id1167339343137\">\n<div id=\"fs-id1167339343138\">\n<p id=\"fs-id1167339343140\">[latex]16{x}^{2}-81[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339343165\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339343166\">[latex]\\left(4x+9\\right)\\left(4x-9\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343214\">\n<div id=\"fs-id1167339343215\">\n<p id=\"fs-id1167339343216\">[latex]{y}^{2}+12y+36[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343246\">\n<div id=\"fs-id1167339343247\">\n<p id=\"fs-id1167339343248\">[latex]27{c}^{3}-1331[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339343274\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339343275\">[latex]\\left(3c-11\\right)\\left(9{c}^{2}+33c+121\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343337\">\n<div id=\"fs-id1167339343338\">\n<p id=\"fs-id1167339343339\">[latex]3x{\\left(x-6\\right)}^{-\\frac{1}{4}}+2{\\left(x-6\\right)}^{\\frac{3}{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339343433\">For the following exercises, simplify the expression.<\/p>\n<div id=\"fs-id1167339343436\">\n<div id=\"fs-id1167339343437\">\n<p id=\"fs-id1167339343438\">[latex]\\frac{2{z}^{2}+7z+3}{{z}^{2}-9}\\cdot \\frac{4{z}^{2}-15z+9}{4{z}^{2}-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339343547\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339343548\">[latex]\\frac{4z-3}{2z-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343585\">\n<div id=\"fs-id1167339343586\">\n<p id=\"fs-id1167339343588\">[latex]\\frac{x}{y}+\\frac{2}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339343617\">\n<div id=\"fs-id1167339343618\">\n<p id=\"fs-id1167339343620\">[latex]\\frac{\\frac{a}{2b}-\\frac{2b}{9a}}{\\frac{3a-2b}{6a}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339343702\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339343703\">[latex]\\frac{3a+2b}{3b}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1167339343742\">\n<dt>least common denominator<\/dt>\n<dd id=\"fs-id1167339343745\">the smallest multiple that two denominators have in common<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339343749\">\n<dt>rational expression<\/dt>\n<dd id=\"fs-id1167339343752\">the quotient of two polynomial 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