Simplifying Expressions By Applying Several Properties
Simplifying Expressions by Applying Several Properties
We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.
Summary of Exponent Properties
If \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\) are real numbers, and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then
\(\begin{array}{ccccc}\mathbf{\text{Product Property}}\hfill & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & \hfill \frac{{a}^{m}}{{b}^{m}}& =\hfill & {a}^{m-n},a\ne 0,m>n\hfill \\ & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & \frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & \hfill {a}^{o}& =\hfill & 1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & \hfill {\left(\frac{a}{b}\right)}^{m}& =\hfill & \frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \end{array}\)
Example
Simplify: \(\frac{{\left({y}^{4}\right)}^{2}}{{y}^{6}}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left({y}^{4}\right)}^{2}}{{y}^{6}}\hfill \\ \text{Multiply the exponents in the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{y}^{8}}{{y}^{6}}\hfill \\ \text{Subtract the exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{y}^{2}\hfill \end{array}\)
Example
Simplify: \(\frac{{b}^{12}}{{\left({b}^{2}\right)}^{6}}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{b}^{12}}{{\left({b}^{2}\right)}^{6}}\hfill \\ \text{Multiply the exponents in the denominator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{b}^{12}}{{b}^{12}}\hfill \\ \text{Subtract the exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{b}^{0}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}\)
Notice that after we simplified the denominator in the first step, the numerator and the denominator were equal. So the final value is equal to 1.
Example
Simplify: \({\left(\frac{{y}^{9}}{{y}^{4}}\right)}^{2}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{{y}^{9}}{{y}^{4}}\right)}^{2}\hfill \\ \begin{array}{c}\text{Remember parentheses come before exponents.}\hfill \\ \text{Notice the bases are the same, so we can simplify}\hfill \\ \text{inside the parentheses. Subtract the exponents.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left({y}^{5}\right)}^{2}\hfill \\ \text{Multiply the exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{y}^{10}\hfill \end{array}\)
Example
Simplify: \({\left(\frac{{j}^{2}}{{k}^{3}}\right)}^{4}.\)
Solution
Here we cannot simplify inside the parentheses first, since the bases are not the same.
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{{j}^{2}}{{k}^{3}}\right)}^{4}\hfill \\ \begin{array}{c}\text{Raise the numerator and denominator to the third power}\hfill \\ \text{using the Quotient to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}.\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left({j}^{2}\right)}^{4}}{{\left({k}^{3}\right)}^{4}}\hfill \\ \text{Use the Power Property and simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{j}^{8}}{{k}^{12}}\hfill \end{array}\)
Example
Simplify: \({\left(\frac{2{m}^{2}}{5n}\right)}^{4}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{2{m}^{2}}{5n}\right)}^{4}\hfill \\ \begin{array}{c}\text{Raise the numerator and denominator to the fourth}\hfill \\ \text{power, using the Quotient to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}.\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(2{m}^{2}\right)}^{4}}{{\left(5n\right)}^{4}}\hfill \\ \text{Raise each factor to the fourth power.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{2}^{4}{\left({m}^{2}\right)}^{4}}{{5}^{4}{n}^{4}}\hfill \\ \text{Use the Power Property and simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{16{m}^{8}}{625{n}^{4}}\hfill \end{array}\)
Example
Simplify: \(\frac{{\left({x}^{3}\right)}^{4}{\left({x}^{2}\right)}^{5}}{{\left({x}^{6}\right)}^{5}}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left({x}^{3}\right)}^{4}{\left({x}^{2}\right)}^{5}}{{\left({x}^{6}\right)}^{5}}\hfill \\ \text{Use the Power Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m·n}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{\left({x}^{12}\right)\left({x}^{10}\right)}{\left({x}^{30}\right)}\hfill \\ \text{Add the exponents in the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{x}^{22}}{{x}^{30}}\hfill \\ \text{Use the Quotient Property,}\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{{x}^{8}}\hfill \end{array}\)
Example
Simplify: \(\frac{{\left(10{p}^{3}\right)}^{2}}{{\left(5p\right)}^{3}{\left(2{p}^{5}\right)}^{4}}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(10{p}^{3}\right)}^{2}}{{\left(5p\right)}^{3}{\left(2{p}^{5}\right)}^{4}}\hfill \\ \text{Use the Product to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(ab\right)}^{m}={a}^{m}{b}^{m}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(10\right)}^{2}{\left({p}^{3}\right)}^{2}}{{\left(5\right)}^{3}{\left(p\right)}^{3}{\left(2\right)}^{4}{\left({p}^{5}\right)}^{4}}\hfill \\ \text{Use the Power Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m·n}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100{p}^{6}}{125{p}^{3}·16{p}^{20}}\hfill \\ \text{Add the exponents in the denominator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100{p}^{6}}{125·16{p}^{23}}\hfill \\ \text{Use the Quotient Property,}\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100}{125·16{p}^{17}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{20{p}^{17}}\hfill \end{array}\)
This lesson is part of:
Polynomials II