# 9.8 Rational exponents  (Page 4/7)

 Page 4 / 7

Simplify: ${\left(16{m}^{\frac{1}{3}}\right)}^{\frac{3}{2}}$ ${\left(81{n}^{\frac{2}{5}}\right)}^{\frac{3}{2}}$ .

$64{m}^{\frac{1}{2}}$ $729{n}^{\frac{3}{5}}$

Simplify: ${\left({m}^{3}{n}^{9}\right)}^{\frac{1}{3}}$ ${\left({p}^{4}{q}^{8}\right)}^{\frac{1}{4}}$ .

## Solution

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

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

We will use both the Product and Quotient Properties in the next example.

Simplify: $\frac{{x}^{\frac{3}{4}}·{x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}}$ $\frac{{y}^{\frac{4}{3}}·y}{{y}^{-\frac{2}{3}}}$ .

## Solution

1. $\begin{array}{cccc}& & & \frac{{x}^{\frac{3}{4}}·{x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Product Property in the numerator,}\hfill \\ \text{add the exponents.}\hfill \end{array}\hfill & & & \frac{{x}^{\frac{2}{4}}}{{x}^{-\frac{6}{4}}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Quotient Property, subtract the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & {x}^{\frac{8}{4}}\hfill \\ \text{Simplify.}\hfill & & & {x}^{2}\hfill \end{array}$

2. $\begin{array}{cccc}& & & \frac{{y}^{\frac{4}{3}}·y}{{y}^{-\frac{2}{3}}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Product Property in the numerator,}\hfill \\ \text{add the exponents.}\hfill \end{array}\hfill & & & \frac{{y}^{\frac{7}{3}}}{{y}^{-\frac{2}{3}}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Quotient Property, subtract the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & {y}^{\frac{9}{3}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & {y}^{3}\hfill \end{array}$

Simplify: $\frac{{m}^{\frac{2}{3}}·{m}^{-\frac{1}{3}}}{{m}^{-\frac{5}{3}}}$ $\frac{{n}^{\frac{1}{6}}·n}{{n}^{-\frac{11}{6}}}$ .

${m}^{2}$ ${n}^{3}$

Simplify: $\frac{{u}^{\frac{4}{5}}·{u}^{-\frac{2}{5}}}{{u}^{-\frac{13}{5}}}$ $\frac{{v}^{\frac{1}{2}}·v}{{v}^{-\frac{7}{2}}}$ .

${u}^{3}$ ${v}^{5}$

## Key concepts

• Summary of Exponent Properties
• If $a,b$ are real numbers and $m,n$ are rational numbers, then
• Product Property ${a}^{m}·{a}^{n}={a}^{m+n}$
• Power Property ${\left({a}^{m}\right)}^{n}={a}^{m·n}$
• Product to a Power ${\left(ab\right)}^{m}={a}^{m}{b}^{m}$
• Quotient Property :
$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}m>n$
$\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}a\ne 0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}n>m$
• Zero Exponent Definition ${a}^{0}=1$ , $a\ne 0$
• Quotient to a Power Property ${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.5em}{0ex}}b\ne 0$

## Practice makes perfect

Simplify Expressions with ${a}^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

${x}^{\frac{1}{2}}$
${y}^{\frac{1}{3}}$
${z}^{\frac{1}{4}}$

${r}^{\frac{1}{2}}$
${s}^{\frac{1}{3}}$
${t}^{\frac{1}{4}}$

$\sqrt{r}$ $\sqrt[3]{s}$ $\sqrt[4]{t}$

${u}^{\frac{1}{5}}$
${v}^{\frac{1}{9}}$
${w}^{\frac{1}{20}}$

${g}^{\frac{1}{7}}$
${h}^{\frac{1}{5}}$
${j}^{\frac{1}{25}}$

$\sqrt[7]{g}$ $\sqrt[5]{h}$ $\sqrt[25]{j}$

In the following exercises, write with a rational exponent.

$-\sqrt[7]{x}$
$\sqrt[9]{y}$
$\sqrt[5]{f}$

$\sqrt[8]{r}$

$\sqrt[4]{t}$

${r}^{\frac{1}{8}}$ ${s}^{\frac{1}{10}}$ ${t}^{\frac{1}{4}}$

$\sqrt[3]{a}$

$\sqrt{c}$

$\sqrt[5]{u}$
$\sqrt{v}$

${u}^{\frac{1}{5}}$ ${v}^{\frac{1}{2}}$ ${w}^{\frac{1}{16}}$

$\sqrt[3]{7c}$
$\sqrt[7]{12d}$
$3\sqrt[4]{5f}$

$\sqrt[4]{5x}$
$\sqrt[8]{9y}$
$7\sqrt[5]{3z}$

${\left(5x\right)}^{\frac{1}{4}}$ ${\left(9y\right)}^{\frac{1}{8}}$ $7{\left(3z\right)}^{\frac{1}{5}}$

$\sqrt{21p}$
$\sqrt[4]{8q}$
$4\sqrt[6]{36r}$

$\sqrt[3]{25a}$
$\sqrt{3b}$

${\left(25a\right)}^{\frac{1}{3}}$ ${\left(3b\right)}^{\frac{1}{2}}$ ${\left(40c\right)}^{\frac{1}{10}}$

In the following exercises, simplify.

${81}^{\frac{1}{2}}$
${125}^{\frac{1}{3}}$
${64}^{\frac{1}{2}}$

${625}^{\frac{1}{4}}$
${243}^{\frac{1}{5}}$
${32}^{\frac{1}{5}}$

5 3 2

${16}^{\frac{1}{4}}$
${16}^{\frac{1}{2}}$
${3125}^{\frac{1}{5}}$

${216}^{\frac{1}{3}}$
${32}^{\frac{1}{5}}$
${81}^{\frac{1}{4}}$

6 2 3

${\left(-216\right)}^{\frac{1}{3}}$
$\text{−}{216}^{\frac{1}{3}}$
${\left(216\right)}^{-\frac{1}{3}}$

${\left(-243\right)}^{\frac{1}{5}}$
$\text{−}{243}^{\frac{1}{5}}$
${\left(243\right)}^{-\frac{1}{5}}$

$-3$ $-3$ $\frac{1}{3}$

${\left(-1\right)}^{\frac{1}{3}}$
${-1}^{\frac{1}{3}}$
${\left(1\right)}^{-\frac{1}{3}}$

${\left(-1000\right)}^{\frac{1}{3}}$
$\text{−}{1000}^{\frac{1}{3}}$
${\left(1000\right)}^{-\frac{1}{3}}$

$-10$ $-10$ $\frac{1}{10}$

${\left(-81\right)}^{\frac{1}{4}}$
$\text{−}{81}^{\frac{1}{4}}$
${\left(81\right)}^{-\frac{1}{4}}$

${\left(-49\right)}^{\frac{1}{2}}$
$\text{−}{49}^{\frac{1}{2}}$
${\left(49\right)}^{-\frac{1}{2}}$

not a real number $-7$ $\frac{1}{7}$

${\left(-36\right)}^{\frac{1}{2}}$
$-{36}^{\frac{1}{2}}$
${\left(36\right)}^{-\frac{1}{2}}$

${\left(-1\right)}^{\frac{1}{4}}$
${\left(1\right)}^{-\frac{1}{4}}$
$\text{−}{1}^{\frac{1}{4}}$

not a real number $1$ $-1$

${\left(-100\right)}^{\frac{1}{2}}$
$\text{−}{100}^{\frac{1}{2}}$
${\left(100\right)}^{-\frac{1}{2}}$

${\left(-32\right)}^{\frac{1}{5}}$
${\left(243\right)}^{-\frac{1}{5}}$
$\text{−}{125}^{\frac{1}{3}}$

$-2$ $\frac{1}{3}$
$-5$

Simplify Expressions with ${a}^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

$\sqrt{{m}^{5}}$
$\sqrt[3]{{n}^{2}}$
$\sqrt[4]{{p}^{3}}$

$\sqrt[4]{{r}^{7}}$
$\sqrt[5]{{s}^{3}}$
$\sqrt[3]{{t}^{7}}$

${r}^{\frac{7}{4}}$ ${s}^{\frac{3}{5}}$ ${t}^{\frac{7}{3}}$

$\sqrt[5]{{u}^{2}}$
$\sqrt[5]{{v}^{8}}$
$\sqrt[9]{{w}^{4}}$

$\sqrt[3]{a}$
$\sqrt{{b}^{5}}$
$\sqrt[3]{{c}^{5}}$

${a}^{\frac{1}{3}}$ ${b}^{\frac{1}{5}}$ ${c}^{\frac{5}{3}}$

In the following exercises, simplify.

${16}^{\frac{3}{2}}$
${8}^{\frac{2}{3}}$
${10,000}^{\frac{3}{4}}$

${1000}^{\frac{2}{3}}$
${25}^{\frac{3}{2}}$
${32}^{\frac{3}{5}}$

100 125 8

${27}^{\frac{5}{3}}$
${16}^{\frac{5}{4}}$
${32}^{\frac{2}{5}}$

${16}^{\frac{3}{2}}$
${125}^{\frac{5}{3}}$
${64}^{\frac{4}{3}}$

64 3125 256

${32}^{\frac{2}{5}}$
${27}^{-\frac{2}{3}}$
${25}^{-\frac{3}{2}}$

${64}^{\frac{5}{2}}$
${81}^{-\frac{3}{2}}$
${27}^{-\frac{4}{3}}$

32,768 $\frac{1}{729}$ $\frac{1}{81}$

${25}^{\frac{3}{2}}$
${9}^{-\frac{3}{2}}$
${\left(-64\right)}^{\frac{2}{3}}$

${100}^{\frac{3}{2}}$
${49}^{-\frac{5}{2}}$
${\left(-100\right)}^{\frac{3}{2}}$

1000 $\frac{1}{16,807}$ not a real number

$\text{−}{9}^{\frac{3}{2}}$
$\text{−}{9}^{-\frac{3}{2}}$
${\left(-9\right)}^{\frac{3}{2}}$

$\text{−}{64}^{\frac{3}{2}}$
$\text{−}{64}^{-\frac{3}{2}}$
${\left(-64\right)}^{\frac{3}{2}}$

$-512$
$-\frac{1}{512}$ not a real number

$\text{−}{100}^{\frac{3}{2}}$
$\text{−}{100}^{-\frac{3}{2}}$
${\left(-100\right)}^{\frac{3}{2}}$

$\text{−}{49}^{\frac{3}{2}}$
$\text{−}{49}^{-\frac{3}{2}}$
${\left(-49\right)}^{\frac{3}{2}}$

$-343$ $-\frac{1}{343}$ not a real number

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${4}^{\frac{5}{8}}·{4}^{\frac{11}{8}}$
${m}^{\frac{7}{12}}·{m}^{\frac{17}{12}}$
${p}^{\frac{3}{7}}·{p}^{\frac{18}{7}}$

${6}^{\frac{5}{2}}·{6}^{\frac{1}{2}}$
${n}^{\frac{2}{10}}·{n}^{\frac{8}{10}}$
${q}^{\frac{2}{5}}·{q}^{\frac{13}{5}}$

216 $n$ ${q}^{3}$

${5}^{\frac{1}{2}}·{5}^{\frac{7}{2}}$
${c}^{\frac{3}{4}}·{c}^{\frac{9}{4}}$
${d}^{\frac{3}{5}}·{d}^{\frac{2}{5}}$

${10}^{\frac{1}{3}}·{10}^{\frac{5}{3}}$
${x}^{\frac{5}{6}}·{x}^{\frac{7}{6}}$
${y}^{\frac{11}{8}}·{y}^{\frac{21}{8}}$

100 ${x}^{2}$ ${y}^{4}$

${\left({m}^{6}\right)}^{\frac{5}{2}}$
${\left({n}^{9}\right)}^{\frac{4}{3}}$
${\left({p}^{12}\right)}^{\frac{3}{4}}$

${\left({a}^{12}\right)}^{\frac{1}{6}}$
${\left({b}^{15}\right)}^{\frac{3}{5}}$
${\left({c}^{11}\right)}^{\frac{1}{11}}$

${a}^{2}$ ${b}^{9}$
$c$

${\left({x}^{12}\right)}^{\frac{2}{3}}$
${\left({y}^{20}\right)}^{\frac{2}{5}}$
${\left({z}^{16}\right)}^{\frac{1}{16}}$

${\left({h}^{6}\right)}^{\frac{4}{3}}$
${\left({k}^{12}\right)}^{\frac{3}{4}}$
${\left({j}^{10}\right)}^{\frac{7}{5}}$

${h}^{8}$ ${k}^{9}$ ${j}^{14}$

$\frac{{x}^{\frac{7}{2}}}{{x}^{\frac{5}{2}}}$
$\frac{{y}^{\frac{5}{2}}}{{y}^{\frac{1}{2}}}$
$\frac{{r}^{\frac{4}{5}}}{{r}^{\frac{9}{5}}}$

$\frac{{s}^{\frac{11}{5}}}{{s}^{\frac{6}{5}}}$
$\frac{{z}^{\frac{7}{3}}}{{z}^{\frac{1}{3}}}$
$\frac{{w}^{\frac{2}{7}}}{{w}^{\frac{9}{7}}}$

$s$ ${z}^{2}$ $\frac{1}{w}$

$\frac{{t}^{\frac{12}{5}}}{{t}^{\frac{7}{5}}}$
$\frac{{x}^{\frac{3}{2}}}{{x}^{\frac{1}{2}}}$
$\frac{{m}^{\frac{13}{8}}}{{m}^{\frac{5}{8}}}$

$\frac{{u}^{\frac{13}{9}}}{{u}^{\frac{4}{9}}}$
$\frac{{r}^{\frac{15}{7}}}{{r}^{\frac{8}{7}}}$
$\frac{{n}^{\frac{3}{5}}}{{n}^{\frac{8}{5}}}$

$u$ $r$ $\frac{1}{n}$

${\left(9{p}^{\frac{2}{3}}\right)}^{\frac{5}{2}}$
${\left(27{q}^{\frac{3}{2}}\right)}^{\frac{4}{3}}$

${\left(81{r}^{\frac{4}{5}}\right)}^{\frac{1}{4}}$
${\left(64{s}^{\frac{3}{7}}\right)}^{\frac{1}{6}}$

$3{r}^{\frac{1}{5}}$ $2{s}^{\frac{1}{14}}$

${\left(16{u}^{\frac{1}{3}}\right)}^{\frac{3}{4}}$
${\left(100{v}^{\frac{2}{5}}\right)}^{\frac{3}{2}}$

${\left(27{m}^{\frac{3}{4}}\right)}^{\frac{2}{3}}$
${\left(625{n}^{\frac{8}{3}}\right)}^{\frac{3}{4}}$

$9{m}^{\frac{1}{2}}$ $125{n}^{2}$

how did you get the value of 2000N.What calculations are needed to arrive at it
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Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
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Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
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