Exponents in Algebra
If you are like most Canadians, your employer pays you biweekly. Assume you earn as follows:
12.00 × (hours worked during the biweekly pay period)
The quantity “hours worked during the biweekly pay period” is the unknown variable. Recall that the word variable 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 to represent “hours worked during the biweekly pay period.” This rewrites the above expression as follows:
or simply
Unfortunately, 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.
For 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:
C7=CCElatex hlatex $12latex 12hlatex hlatex 12hlatex 12\cdot hlatex 12latex hlatex =latex 6x+3ylatex 4x+3latex xlatex ylatex 6x+3y=4x+3latex xlatex ylatex xlatex ylatex 6x + 3ylatex 6xlatex 3ylatex 12hlatex 6x + 3ylatex 6xlatex 6latex xlatex 6xlatex \frac{7x}{3}+4xy^2=x^3-2ylatex \frac{7x}{3}latex 4xy^2latex \frac{7x}{3}latex \frac{7}{3}xlatex \frac{7}{3}latex xlatex \frac{7}{3}latex xlatex x^3latex 2ylatex x^3latex xlatex xlatex xlatex 1\cdot x^3latex -2ylatex -2latex ylatex n latex 1, 2, 3, \ldots latex blatex b^n
latex 2^3 = 8latex 2 × 2 × 2latex y^ay^b = y^{a+b}latex \dfrac{y^a}{y^b} = y^{a-b}latex (x^a)^b = x^{ab}latex (xy)^a=x^ay^alatex \left(\dfrac{x}{y}\right)^a=\dfrac{x^a}{y^a}latex y^0 = 1latex y^{−a} = \frac{1}{y^a}latex y^{\frac{a}{b} }= \sqrt[b]{y^a}latex 1latex 1latex 2latex 2^1latex 2latex (yz)^alatex (y^1z^1)^alatex (y^{1a}z^{1a})latex y^az^alatex 1
latex 7q + 4q + 14qlatex qlatex 25qlatex 7q + 4q + 14q
latex 7qlatex 4qlatex qlatex 7qlatex 4q^2latex qlatex q^2latex 1latex xlatex 1xlatex \frac{x}{4}latex \frac{{1x}}{4}latex \frac{1}{4}xlatex 7x^0latex 7(1)latex 7latex \frac{3}{2}x + 4{x^2} – 10x – 2y + \frac{x}{3}latex 23{g^2} – \frac{{17{g^2}}}{5} + {g^4} + {g^2} – \frac{2}{3}{g^2} – 0.15g + {g^3}latex xlatex g^2
latex 2(4a + 6b) − (2a − 3b)latex 2(4a + 6b) + (−1)(2a − 3b)latex −(2a − 3b)latex −2a + 3blatex 2(4a + 6b) − 2a + 3b
latex 2 − a + 3blatex \frac{{4a}}{{4a}}latex \frac{{4a}}{{4a}} = 0latex \frac{4}{4} = 1latex \frac{a}{a} = 1latex \frac{{4a}}{{4a}} = \left( 1 \right)\left( 1 \right) = 1latex \frac{{{a^1}}}{{{a^1}}} = {a^{1 – 1}} = {a^0} = 1
latex {PV} = \frac{FV}{{1+rt}}latex FV = $5,443.84latex r = 0.12latex t = \frac{270}{365}
latex FVlatex Flatex Vlatex d_1latex d_2latex 2x + 5xlatex 7xlatex x = 2latex 2x + 5xlatex 2(2) + 5(2) = 14latex 7(2) = 14latex 2a − 3a + 4 + 6a − 3latex 5b(4b + 2)latex \frac{{6{x^3}+12{x^2}+13.5x}}{{3x}}latex {(1 + i)^3} \cdot {(1 + i)^{14}}latex {8^{\frac{2}{3}}}latex I = Prtlatex P = $2,500, r = 0.06latex (6r^2+10-6r+4r^2-3)-(-12r-5r^2+2+3r)latex \left[ \dfrac{5x^9 + 3x^9}{2x} \right]^5latex \dfrac{t}{2} + 0.75t – {t^3} + \dfrac{{5{t^4}}}{t} – \dfrac{{2\left( {t + {t^3}} \right)}}{4}latex \dfrac{14\left( 1+i \right)+21\left( 1+i\right)^4 – 35\left( 1+i\right)^7}{7\left(1+i \right)}latex \dfrac{R}{1+0.08\cdot \frac{183}{365}} + 3R\left( 1 + 0.08 \cdot\frac{52}{365} \right)latex \left[ \left( \dfrac{2}{5}\right)^2\right]^2latex PV = \dfrac{FV}{\left( 1+i\right)^N}latex FV = $3,417.24latex i = 0.05latex N = 6latex FV = $10,000, N = 17latex i = 0.10latex \left[\dfrac{10a^2b^3c^4}{5b^3c^4}\right]^2 + 6\left(a^8\right)^{1/2} – \left( 3a^2 + 6\right)\left(3a^2 – 3\right)latex \dfrac{- \left(5x+4y+3\right)\left(2x-5y\right)-\left(10x-2y\right)\left(2y+3\right)}{5}latex \dfrac{\left(-3z \right)^3}{\left(3z^2 \right)^2}{\left(2z^3\right)}^{ – 4}
latex PMT = $500latex i = 0.05latex Δ\% = 0.02latex CY = 2latex PY = 4latex N = 20latex 50,000\cdot \left(1 + \dfrac{0.10}{12}\right)^{ – 27}latex 995\left[\dfrac{1-\left(1+0.02\right)^{13}\left(1+\frac{0.09}{4}\right)^{ – 13}}{\frac{0.09}{4}-0.02}\right]$