{"id":191,"date":"2021-10-20T22:57:45","date_gmt":"2021-10-20T22:57:45","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/mathtest\/?post_type=chapter&p=191"},"modified":"2023-06-14T16:21:43","modified_gmt":"2023-06-14T16:21:43","slug":"exponents-in-algebra","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/mathtest\/chapter\/exponents-in-algebra\/","title":{"raw":"Exponents in Algebra","rendered":"Exponents in Algebra"},"content":{"raw":"If you are like most Canadians, your employer pays you biweekly. Assume you earn $12.00 per hour. How do you calculate your pay cheque every pay period? Your earnings are calculated in $ as follows:\r\n

12.00 \u00d7 (hours worked during the biweekly pay period)<\/p>\r\nThe quantity \"hours worked during the biweekly pay period\" is the unknown variable. Recall that the word variable<\/strong> simply represents a quantity that can vary in value. Notice that the expression above appears lengthy when you write out the explanation for the variable. Algebra uses symbols to make such expressions more convenient to manipulate. To shorten the expression, making it easier to read, we can assign a letter or a group of letters to represent the variable. In this case, you might choose $latex h$ to represent \"hours worked during the biweekly pay period.\" This rewrites the above expression as follows:\r\n

[latex]12.00 \\times h[\/latex] or simply\u00a0 [latex]12h[\/latex]<\/p>\r\nUnfortunately, the word algebra makes many people worry. But remember that algebra is just a set of tools that can help in solving a numerical problem. It is used to demonstrate how the pieces of a puzzle fit together to arrive at a solution.\r\n\r\nFor example, you have used your algebraic skills if you have ever programmed a formula into Microsoft Excel. You told Excel there was a relationship between cells in your spreadsheet. Perhaps your calculation required cell C3 to be divided by the sum of cells C3, D3 and E3, and then multiplied by cell B7. This is an algebraic expression. Excel then took your algebraic expression and calculated the solution by automatically substituting in the appropriate values from the referenced cells (your variables). See the example below:\r\n\r\nC7=C$3\/SUM($C$3:$E$3)*B7\r\n\r\n\"Screenshot\r\n\r\nAlgebra involves integrating many interrelated concepts. In this textbook we will discuss the concepts that are important to business mathematics. Your understanding of algebra will become more complete as more concepts are covered over the course of this book.\r\n\r\nThis section reviews the language of algebra, including exponent rules, basic operation rules, and substitution. In Section 1.2 you will put these concepts to work in solving one linear equation for one unknown variable along with two linear equations with two unknown variables. Finally, in Section 1.3 you will explore the concepts of logarithms and natural logarithms.\r\n

The Language of Algebra<\/h2>\r\nUnderstanding the rules of algebra requires familiarity with four key definitions.\r\n

Algebraic Expressions<\/h3>\r\nA mathematical algebraic expression indicates the relationship between the specific values represented by numbers or variables and the mathematical operations that must be conducted on these values. For example, if $latex h$ represents the number of hours worked and the hourly wage is $latex \\$12$, the expression $latex 12h$ calculates the total amount of dollars earned for $latex h$ hours worked. Note that, even though by convention we don't write a multiplication symbol in this expression, $latex 12h$ represents $latex 12\\cdot h$. Specifically, $latex 12$ must be multiplied by the value of $latex h$.\r\n\r\nNote also that an expression does not include an equal sign, or \"$latex =$\". An expression only tells you what to do and requires that you substitute a value in for the unknown variable(s) to solve. There is no one definable solution to the expression - an expression can represent many different values, potentially infinitely many. The value of the expression will change as the value of the variables in the expression change. An algebraic expression is not the same thing as <\/strong>an algebraic equation, however an algebraic equation is built from algebraic expressions.\r\n

Algebraic Equations<\/h3>\r\nAn algebraic equation<\/strong> is a statement that says two algebraic expressions are equal. This equation can often, but not always, be solved to find a solution for the unknown variables.\r\n\r\nExamine the following illustration to see how algebraic expressions and algebraic equations are interrelated.\r\n\r\nConsider two algebraic expressions:\r\n

$latex 6x+3y$ and $latex 4x+3$<\/p>\r\nEach of these expressions represents a value that can be calculated if we were given values for $latex x$ and $latex y$.\r\n\r\nBy writing\r\n

$latex 6x+3y=4x+3$<\/p>\r\nwe are imposing a condition on these two expressions, stating that they must produce equal values. This in turn imposes a condition on the values of $latex x$ and $latex y$; we can potentially no longer substitute any value for $latex x$ and $latex y$ because we may end up with a statement that is not true.\r\n

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Concept Check<\/strong><\/h4>\r\nCan you determine whether a particular value or a set of values is a solution to a given equation? Try it out:<\/span>\r\n\r\nMathMatize: Verifying solutions to equations<\/a>\r\n\r\n<\/div>\r\n

Terms<\/h3>\r\nIn any algebraic expression, terms are the components that are separated by addition and subtraction. In looking at the example above, the expression $latex 6x + 3y$ is composed of two terms. These terms are $latex 6x$ and $latex 3y$. A nomial<\/strong> refers to how many terms appear in an algebraic expression. If an algebraic expression contains only one term, like $latex 12h$, it is called a monomial<\/strong> (mono = one). If the expression contains two terms or more, such as $latex 6x + 3y$, it is called a polynomial<\/strong> (poly = many).\r\n

Factors<\/h3>\r\nTerms may consist of one or more factors<\/strong> that are separated by multiplication or division signs. For example, $latex 6x$ consists of two factors. These factors are $latex 6$ and $latex x$; they are joined by multiplication to create $latex 6x$.\r\n