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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
5%
Michael
hey everyone how to do algebra
Felecia answer 1.5 hours before he reaches her
I would like to solve the problem -6/2x
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year?
Tickets for the community fair cost $12 for adults and$5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold? Alpha Reply 220 gayla Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert? Tsimmuaj Reply Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj ? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year candido what would be wong candido 87 divided by 3 then multiply that by 4. 116 people total. Melissa the actual number that has 3 out of 4 of a whole pie candido was having a hard time finding Teddy use Matrices for the 2nd question Daniel One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers. Tsimmuaj Reply 26 + 37 = 63 + 8 = 71 gayla 26+37=63+8=71 ziad 11+52=63+8=71 Thisha how do we know the answer is correct? Thisha 23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for. gayla 23 +11 = 37. 23+37=63 63+8=71 Gayla by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71 Gayla your answer did not fit the guidelines of the question 11 is 41 less than 52. gayla 71-8-11 =52 is this correct? Ruel let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer tesfu just new Muwanga Amara currently sells televisions for company A at a salary of$17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of$29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal? Tsimmuaj Reply yes math Kenneth company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100 gayla need help with math to do tsi test Toocute me too Christian have you tried the TSI practice test ***tsipracticetest.com gayla DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players. gustavo Reply ? Ann DaMarcus: 16 mi/hr Fabian: 22 mi/hr Sherman Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution? Wenda Reply 15 and 5 32 is 40% , & 8 is 10 % , & any 4 letters is 5%. Karen It felt that something is missing on the question like: 40% of what solution? 10% of what solution? Jhea its confusing Sparcast 3% & 2% to complete the 25% Sparcast because she already has 20 liters. Sparcast ok I was a little confused I agree 15% & 5% Sparcast 8,2 Karen Jim and Debbie earned$7200. Debbie earned \$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike