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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

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(10{p}^{3}\right)}^{2}}{{\left(5p\right)}^{3}{\left(2{p}^{5}\right)}^{4}}\hfill \\ \text{Use the Product to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(ab\right)}^{m}={a}^{m}{b}^{m}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(10\right)}^{2}{\left({p}^{3}\right)}^{2}}{{\left(5\right)}^{3}{\left(p\right)}^{3}{\left(2\right)}^{4}{\left({p}^{5}\right)}^{4}}\hfill \\ \text{Use the Power Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m·n}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100{p}^{6}}{125{p}^{3}·16{p}^{20}}\hfill \\ \text{Add the exponents in the denominator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100{p}^{6}}{125·16{p}^{23}}\hfill \\ \text{Use the Quotient Property,}\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100}{125·16{p}^{17}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{20{p}^{17}}\hfill \end{array}$$

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