# 9.8 Rational exponents  (Page 3/7)

 Page 3 / 7

Simplify: ${8}^{-\frac{5}{3}}$ ${81}^{-\frac{3}{2}}$ ${16}^{-\frac{3}{4}}$ .

$\frac{1}{32}$ $\frac{1}{729}$ $\frac{1}{8}$

Simplify: ${4}^{-\frac{3}{2}}$ ${27}^{-\frac{2}{3}}$ ${625}^{-\frac{3}{4}}$ .

$\frac{1}{8}$ $\frac{1}{9}$ $\frac{1}{125}$

Simplify: $\text{−}{25}^{\frac{3}{2}}$ $\text{−}{25}^{-\frac{3}{2}}$ ${\left(-25\right)}^{\frac{3}{2}}$ .

## Solution

$\begin{array}{cccc}& & & \phantom{\rule{2em}{0ex}}\text{−}{25}^{\frac{3}{2}}\hfill \\ \\ \\ \text{Rewrite in radical form.}\hfill & & & \phantom{\rule{2em}{0ex}}\text{−}{\left(\sqrt{25}\right)}^{3}\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & \phantom{\rule{2em}{0ex}}\text{−}{\left(5\right)}^{3}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \phantom{\rule{2em}{0ex}}-125\hfill \end{array}$

$\begin{array}{cccc}& & & \phantom{\rule{2em}{0ex}}\text{−}{25}^{-\frac{3}{2}}\hfill \\ \\ \\ \text{Rewrite using}\phantom{\rule{0.2em}{0ex}}{b}^{\text{−}p}=\frac{1}{{b}^{p}}.\hfill & & & \phantom{\rule{2em}{0ex}}\text{−}\left(\frac{1}{{25}^{\frac{3}{2}}}\right)\hfill \\ \\ \\ \text{Rewrite in radical form.}\hfill & & & \phantom{\rule{2em}{0ex}}\text{−}\left(\frac{1}{{\left(\sqrt{25}\right)}^{3}}\right)\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & \phantom{\rule{2em}{0ex}}\text{−}\left(\frac{1}{{\left(5\right)}^{3}}\right)\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \phantom{\rule{2em}{0ex}}-\frac{1}{125}\hfill \end{array}$

$\begin{array}{cccc}& & & {\left(-25\right)}^{\frac{3}{2}}\hfill \\ \text{Rewrite in radical form.}\hfill & & & {\left(\sqrt{-25}\right)}^{3}\hfill \\ \begin{array}{c}\text{There is no real number whose}\hfill \\ \text{square root is}\phantom{\rule{0.2em}{0ex}}-25.\hfill \end{array}\hfill & & & \text{Not a real number.}\hfill \end{array}$

Simplify: ${-16}^{\frac{3}{2}}$ ${-16}^{-\frac{3}{2}}$ ${\left(-16\right)}^{-\frac{3}{2}}$ .

$-64$ $-\frac{1}{64}$ not a real number

Simplify: ${-81}^{\frac{3}{2}}$ ${-81}^{-\frac{3}{2}}$ ${\left(-81\right)}^{-\frac{3}{2}}$ .

$-729$ $-\frac{1}{729}$ not a real number

## Use the laws of exponents to simplify expressions with rational exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

## Summary of exponent properties

If $a,b$ are real numbers and $m,n$ are rational numbers, then

$\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}·{a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}m>n\hfill \\ & & & \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & & {a}^{0}=1,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}b\ne 0\hfill \end{array}$

When we multiply the same base, we add the exponents.

Simplify: ${2}^{\frac{1}{2}}·{2}^{\frac{5}{2}}$ ${x}^{\frac{2}{3}}·{x}^{\frac{4}{3}}$ ${z}^{\frac{3}{4}}·{z}^{\frac{5}{4}}$ .

## Solution

$\begin{array}{cccc}& & & {2}^{\frac{1}{2}}·{2}^{\frac{5}{2}}\hfill \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & {2}^{\frac{1}{2}+\frac{5}{2}}\hfill \\ \text{Add the fractions.}\hfill & & & {2}^{\frac{6}{2}}\hfill \\ \text{Simplify the exponent.}\hfill & & & {2}^{3}\hfill \\ \text{Simplify.}\hfill & & & 8\hfill \end{array}$

$\begin{array}{cccc}& & & {x}^{\frac{2}{3}}·{x}^{\frac{4}{3}}\hfill \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & {x}^{\frac{2}{3}+\frac{4}{3}}\hfill \\ \text{Add the fractions.}\hfill & & & {x}^{\frac{6}{3}}\hfill \\ \text{Simplify.}\hfill & & & {x}^{2}\hfill \end{array}$

$\begin{array}{cccc}& & & {z}^{\frac{3}{4}}·{z}^{\frac{5}{4}}\hfill \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & {z}^{\frac{3}{4}+\frac{5}{4}}\hfill \\ \text{Add the fractions.}\hfill & & & {z}^{\frac{8}{4}}\hfill \\ \text{Simplify.}\hfill & & & {z}^{2}\hfill \end{array}$

Simplify: ${3}^{\frac{2}{3}}·{3}^{\frac{4}{3}}$ ${y}^{\frac{1}{3}}·{y}^{\frac{8}{3}}$ ${m}^{\frac{1}{4}}·{m}^{\frac{3}{4}}$ .

9 ${y}^{3}$ m

Simplify: ${5}^{\frac{3}{5}}·{5}^{\frac{7}{5}}$ ${z}^{\frac{1}{8}}·{z}^{\frac{7}{8}}$ ${n}^{\frac{2}{7}}·{n}^{\frac{5}{7}}$ .

25 z n

We will use the Power Property in the next example.

Simplify: ${\left({x}^{4}\right)}^{\frac{1}{2}}$ ${\left({y}^{6}\right)}^{\frac{1}{3}}$ ${\left({z}^{9}\right)}^{\frac{2}{3}}$ .

## Solution

$\begin{array}{cccc}& & & {\left({x}^{4}\right)}^{\frac{1}{2}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & {x}^{4·\frac{1}{2}}\hfill \\ \text{Simplify.}\hfill & & & {x}^{2}\hfill \end{array}$

$\begin{array}{cccc}& & & {\left({y}^{6}\right)}^{\frac{1}{3}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & {y}^{6·\frac{1}{3}}\hfill \\ \text{Simplify.}\hfill & & & {y}^{2}\hfill \end{array}$

$\begin{array}{cccc}& & & {\left({z}^{9}\right)}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & {z}^{9·\frac{2}{3}}\hfill \\ \text{Simplify.}\hfill & & & {z}^{6}\hfill \end{array}$

Simplify: ${\left({p}^{10}\right)}^{\frac{1}{5}}$ ${\left({q}^{8}\right)}^{\frac{3}{4}}$ ${\left({x}^{6}\right)}^{\frac{4}{3}}$ .

${p}^{2}$ ${q}^{6}$ ${x}^{8}$

Simplify: ${\left({r}^{6}\right)}^{\frac{5}{3}}$ ${\left({s}^{12}\right)}^{\frac{3}{4}}$ ${\left({m}^{9}\right)}^{\frac{2}{9}}$ .

${r}^{10}$ ${s}^{9}$ ${m}^{2}$

The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

Simplify: $\frac{{x}^{\frac{4}{3}}}{{x}^{\frac{1}{3}}}$ $\frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}}$ $\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}$ .

## Solution

1. $\begin{array}{cccc}& & & \frac{{x}^{\frac{4}{3}}}{{x}^{\frac{1}{3}}}\hfill \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & {x}^{\frac{4}{3}-\frac{1}{3}}\hfill \\ \text{Simplify.}\hfill & & & x\hfill \end{array}$

2. $\begin{array}{cccc}& & & \frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}}\hfill \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & {y}^{\frac{3}{4}-\frac{1}{4}}\hfill \\ \text{Simplify.}\hfill & & & {y}^{\frac{1}{2}}\hfill \end{array}$

3. $\begin{array}{cccc}& & & \frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}\hfill \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & {z}^{\frac{2}{3}-\frac{5}{3}}\hfill \\ \text{Rewrite without a negative exponent.}\hfill & & & \frac{1}{z}\hfill \end{array}$

Simplify: $\frac{{u}^{\frac{5}{4}}}{{u}^{\frac{1}{4}}}$ $\frac{{v}^{\frac{3}{5}}}{{v}^{\frac{2}{5}}}$ $\frac{{x}^{\frac{2}{3}}}{{x}^{\frac{5}{3}}}$ .

u ${v}^{\frac{1}{5}}$ $\frac{1}{x}$

Simplify: $\frac{{c}^{\frac{12}{5}}}{{c}^{\frac{2}{5}}}$ $\frac{{m}^{\frac{5}{4}}}{{m}^{\frac{9}{4}}}$ $\frac{{d}^{\frac{1}{5}}}{{d}^{\frac{6}{5}}}$ .

${c}^{6}$ $\frac{1}{m}$ $\frac{1}{d}$

Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.

Simplify: ${\left(27{u}^{\frac{1}{2}}\right)}^{\frac{2}{3}}$ ${\left(8{v}^{\frac{1}{4}}\right)}^{\frac{2}{3}}$ .

## Solution

1. $\begin{array}{cccc}& & & {\left(27{u}^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{First we use the Product to a Power}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & {\left(27\right)}^{\frac{2}{3}}{\left({u}^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \text{Rewrite 27 as a power of 3.}\hfill & & & {\left({3}^{3}\right)}^{\frac{2}{3}}{\left({u}^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \left({3}^{2}\right)\left({u}^{\frac{1}{3}}\right)\hfill \\ \\ \\ \text{Simplify.}\hfill & & & 9{u}^{\frac{1}{3}}\hfill \end{array}$

2. $\begin{array}{cccc}& & & {\left(8{v}^{\frac{1}{4}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{First we use the Product to a Power}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & {\left(8\right)}^{\frac{2}{3}}{\left({v}^{\frac{1}{4}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \text{Rewrite 8 as a power of 2.}\hfill & & & {\left({2}^{3}\right)}^{\frac{2}{3}}{\left({v}^{\frac{1}{4}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \left({2}^{2}\right)\left({v}^{\frac{1}{6}}\right)\hfill \\ \\ \\ \text{Simplify.}\hfill & & & 4{v}^{\frac{1}{6}}\hfill \end{array}$

Simplify: ${\left(32{x}^{\frac{1}{3}}\right)}^{\frac{3}{5}}$ ${\left(64{y}^{\frac{2}{3}}\right)}^{\frac{1}{3}}$ .

$8{x}^{\frac{1}{5}}$ $4{y}^{\frac{2}{9}}$

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