# 9.8 Rational exponents  (Page 5/7)

 Page 5 / 7

${\left({x}^{8}{y}^{10}\right)}^{\frac{1}{2}}$
${\left({a}^{9}{b}^{12}\right)}^{\frac{1}{3}}$

${\left({r}^{8}{s}^{4}\right)}^{\frac{1}{4}}$
${\left({u}^{15}{v}^{20}\right)}^{\frac{1}{5}}$

${r}^{2}s$ ${u}^{3}{v}^{4}$

${\left({a}^{6}{b}^{16}\right)}^{\frac{1}{2}}$
${\left({j}^{9}{k}^{6}\right)}^{\frac{2}{3}}$

${\left({r}^{16}{s}^{10}\right)}^{\frac{1}{2}}$
${\left({u}^{10}{v}^{5}\right)}^{\frac{4}{5}}$

${r}^{8}{s}^{5}$ ${u}^{8}{v}^{4}$

$\frac{{r}^{\frac{5}{2}}·{r}^{-\frac{1}{2}}}{{r}^{-\frac{3}{2}}}$
$\frac{{s}^{\frac{1}{5}}·s}{{s}^{-\frac{9}{5}}}$

$\frac{{a}^{\frac{3}{4}}·{a}^{-\frac{1}{4}}}{{a}^{-\frac{10}{4}}}$
$\frac{{b}^{\frac{2}{3}}·b}{{b}^{-\frac{7}{3}}}$

${a}^{3}$ ${b}^{4}$

$\frac{{c}^{\frac{5}{3}}·{c}^{-\frac{1}{3}}}{{c}^{-\frac{2}{3}}}$
$\frac{{d}^{\frac{3}{5}}·d}{{d}^{-\frac{2}{5}}}$

$\frac{{m}^{\frac{7}{4}}·{m}^{-\frac{5}{4}}}{{m}^{-\frac{2}{4}}}$
$\frac{{n}^{\frac{3}{7}}·n}{{n}^{-\frac{4}{7}}}$

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

${4}^{\frac{5}{2}}·{4}^{\frac{1}{2}}$

${n}^{\frac{2}{6}}·{n}^{\frac{4}{6}}$

$n$

${\left({a}^{24}\right)}^{\frac{1}{6}}$

${\left({b}^{10}\right)}^{\frac{3}{5}}$

${b}^{6}$

$\frac{{w}^{\frac{2}{5}}}{{w}^{\frac{7}{5}}}$

$\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{8}{3}}}$

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

${\left(27{r}^{\frac{3}{5}}\right)}^{\frac{1}{3}}$

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

$2{s}^{\frac{1}{10}}$

${\left({r}^{9}{s}^{12}\right)}^{\frac{1}{3}}$

${\left({u}^{12}{v}^{18}\right)}^{\frac{1}{6}}$

${u}^{2}{v}^{3}$

## Everyday math

Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify ${144}^{\frac{1}{2}}$ to find the length of each side of his garden.

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify ${242}^{\frac{1}{2}}$ to find the length of each side of his patio.Round to the nearest tenth of a foot.

15.6 feet

Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify $\frac{{12}^{\frac{1}{2}}}{{16}^{\frac{1}{2}}}$ to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify $\frac{{1024}^{\frac{1}{2}}}{{16}^{\frac{1}{2}}}$ to find how many seconds it took for the flare to reach the water.

8 seconds

## Writing exercises

Show two different algebraic methods to simplify ${4}^{\frac{3}{2}}.$ Explain all your steps.

Explain why the expression ${\left(-16\right)}^{\frac{3}{2}}$ cannot be evaluated.

## Section 9.1 Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{64}$

$\sqrt{144}$

12

$-\sqrt{25}$

$-\sqrt{81}$

$-9$

$\sqrt{-9}$

$\sqrt{-36}$

not a real number

$\sqrt{64}+\sqrt{225}$

$\sqrt{64+225}$

17

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

$\sqrt{28}$

$\sqrt{155}$

$12<\sqrt{155}<13$

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

$\sqrt{15}$

$\sqrt{57}$

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{{q}^{2}}$

$\sqrt{64{b}^{2}}$

$8b$

$\text{−}\sqrt{121{a}^{2}}$

$\sqrt{225{m}^{2}{n}^{2}}$

$15mn$

$\text{−}\sqrt{100{q}^{2}}$

$\sqrt{49{y}^{2}}$

$7y$

$\sqrt{4{a}^{2}{b}^{2}}$

$\sqrt{121{c}^{2}{d}^{2}}$

$11cd$

## Section 9.2 Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{300}$

$\sqrt{98}$

$7\sqrt{2}$

$\sqrt{{x}^{13}}$

$\sqrt{{y}^{19}}$

${y}^{9}\sqrt{y}$

$\sqrt{16{m}^{4}}$

$\sqrt{36{n}^{13}}$

$6{n}^{6}\sqrt{n}$

$\sqrt{288{m}^{21}}$

$\sqrt{150{n}^{7}}$

$5{n}^{3}\sqrt{6n}$

$\sqrt{48{r}^{5}{s}^{4}}$

$\sqrt{108{r}^{5}{s}^{3}}$

$6{r}^{2}s\sqrt{3rs}$

$\frac{10-\sqrt{50}}{5}$

$\frac{6+\sqrt{72}}{6}$

$1+\sqrt{2}$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{16}{25}}$

$\sqrt{\frac{81}{36}}$

$\frac{3}{2}$

$\sqrt{\frac{{x}^{8}}{{x}^{4}}}$

$\sqrt{\frac{{y}^{6}}{{y}^{2}}}$

${y}^{2}$

$\sqrt{\frac{98{p}^{6}}{2{p}^{2}}}$

$\sqrt{\frac{72{q}^{8}}{2{q}^{4}}}$

$6{q}^{2}$

$\sqrt{\frac{65}{121}}$

$\sqrt{\frac{26}{169}}$

$\frac{\sqrt{26}}{13}$

$\sqrt{\frac{64{x}^{4}}{25{x}^{2}}}$

$\sqrt{\frac{36{r}^{10}}{16{r}^{5}}}$

$\frac{3{r}^{2}\sqrt{r}}{2}$

$\sqrt{\frac{48{p}^{3}{q}^{5}}{27pq}}$

$\sqrt{\frac{12{r}^{5}{s}^{7}}{75{r}^{2}s}}$

$\frac{2r{s}^{3}\sqrt{r}}{5}$

## Section 9.3 Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

$3\sqrt{2}+\sqrt{2}$

$5\sqrt{5}+7\sqrt{5}$

$12\sqrt{5}$

$4\sqrt{y}+4\sqrt{y}$

$6\sqrt{m}-2\sqrt{m}$

$4\sqrt{m}$

$-3\sqrt{7}+2\sqrt{7}-\sqrt{7}$

$8\sqrt{13}+2\sqrt{3}+3\sqrt{13}$

$11\sqrt{13}+2\sqrt{3}$

$3\sqrt{5xy}-\sqrt{5xy}+3\sqrt{5xy}$

$2\sqrt{3rs}+\sqrt{3rs}-5\sqrt{rs}$

$3\sqrt{3rs}-5\sqrt{rs}$

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{32}+3\sqrt{2}$

$\sqrt{8}+3\sqrt{2}$

$5\sqrt{2}$

$\sqrt{72}+\sqrt{50}$

$\sqrt{48}+\sqrt{75}$

$9\sqrt{3}$

$3\sqrt{32}+\sqrt{98}$

$\frac{1}{3}\sqrt{27}-\frac{1}{8}\sqrt{192}$

$0$

$\sqrt{50{y}^{5}}-\sqrt{72{y}^{5}}$

$6\sqrt{18{n}^{4}}-3\sqrt{8{n}^{4}}+{n}^{2}\sqrt{50}$

$17{n}^{2}\sqrt{2}$

## Section 9.4 Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

$\sqrt{2}·\sqrt{20}$

$2\sqrt{2}·6\sqrt{14}$

$24\sqrt{7}$

$\sqrt{2{m}^{2}}·\sqrt{20{m}^{4}}$

$\left(6\sqrt{2y}\right)\left(3\sqrt{50{y}^{3}}\right)$

$180{y}^{2}$

$\left(6\sqrt{3{v}^{4}}\right)\left(5\sqrt{30v}\right)$

${\left(\sqrt{8}\right)}^{2}$

8

${\left(\text{−}\sqrt{10}\right)}^{2}$

$\left(2\sqrt{5}\right)\left(5\sqrt{5}\right)$

50

$\left(-3\sqrt{3}\right)\left(5\sqrt{18}\right)$

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$10\left(2-\sqrt{7}\right)$

$20-10\sqrt{7}$

$\sqrt{3}\left(4+\sqrt{12}\right)$

$\left(5+\sqrt{2}\right)\left(3-\sqrt{2}\right)$

$13-2\sqrt{2}$

$\left(5-3\sqrt{7}\right)\left(1-2\sqrt{7}\right)$

$\left(1-3\sqrt{x}\right)\left(5+2\sqrt{x}\right)$

$5-13\sqrt{x}-6x$

$\left(3+4\sqrt{y}\right)\left(10-\sqrt{y}\right)$

${\left(1+6\sqrt{p}\right)}^{2}$

$1+12\sqrt{p}+36p$

${\left(2-6\sqrt{5}\right)}^{2}$

$\left(3+2\sqrt{7}\right)\left(3-2\sqrt{7}\right)$

$-19$

$\left(6-\sqrt{11}\right)\left(6+\sqrt{11}\right)$

## Section 9.5 Divide Square Roots

Divide Square Roots

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
x-3y =1; 3x-2y+4=0 graph
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