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ⓐ
${\left({x}^{8}{y}^{10}\right)}^{\frac{1}{2}}$
ⓑ
${\left({a}^{9}{b}^{12}\right)}^{\frac{1}{3}}$
ⓐ
${\left({r}^{8}{s}^{4}\right)}^{\frac{1}{4}}$
ⓑ
${\left({u}^{15}{v}^{20}\right)}^{\frac{1}{5}}$
ⓐ ${r}^{2}s$ ⓑ ${u}^{3}{v}^{4}$
ⓐ
${\left({a}^{6}{b}^{16}\right)}^{\frac{1}{2}}$
ⓑ
${\left({j}^{9}{k}^{6}\right)}^{\frac{2}{3}}$
ⓐ
${\left({r}^{16}{s}^{10}\right)}^{\frac{1}{2}}$
ⓑ
${\left({u}^{10}{v}^{5}\right)}^{\frac{4}{5}}$
ⓐ ${r}^{8}{s}^{5}$ ⓑ ${u}^{8}{v}^{4}$
ⓐ
$\frac{{r}^{\frac{5}{2}}\xb7{r}^{-\frac{1}{2}}}{{r}^{-\frac{3}{2}}}$
ⓑ
$\frac{{s}^{\frac{1}{5}}\xb7s}{{s}^{-\frac{9}{5}}}$
ⓐ
$\frac{{a}^{\frac{3}{4}}\xb7{a}^{-\frac{1}{4}}}{{a}^{-\frac{10}{4}}}$
ⓑ
$\frac{{b}^{\frac{2}{3}}\xb7b}{{b}^{-\frac{7}{3}}}$
ⓐ ${a}^{3}$ ⓑ ${b}^{4}$
ⓐ
$\frac{{c}^{\frac{5}{3}}\xb7{c}^{-\frac{1}{3}}}{{c}^{-\frac{2}{3}}}$
ⓑ
$\frac{{d}^{\frac{3}{5}}\xb7d}{{d}^{-\frac{2}{5}}}$
ⓐ
$\frac{{m}^{\frac{7}{4}}\xb7{m}^{-\frac{5}{4}}}{{m}^{-\frac{2}{4}}}$
ⓑ
$\frac{{n}^{\frac{3}{7}}\xb7n}{{n}^{-\frac{4}{7}}}$
ⓐ $m$ ⓑ ${n}^{2}$
${4}^{\frac{5}{2}}\xb7{4}^{\frac{1}{2}}$
${\left({a}^{24}\right)}^{\frac{1}{6}}$
$\frac{{w}^{\frac{2}{5}}}{{w}^{\frac{7}{5}}}$
$\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{8}{3}}}$
$\frac{1}{{z}^{2}}$
${\left(27{r}^{\frac{3}{5}}\right)}^{\frac{1}{3}}$
${\left(64{s}^{\frac{3}{5}}\right)}^{\frac{1}{6}}$
$2{s}^{\frac{1}{10}}$
${\left({r}^{9}{s}^{12}\right)}^{\frac{1}{3}}$
${\left({u}^{12}{v}^{18}\right)}^{\frac{1}{6}}$
${u}^{2}{v}^{3}$
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
Show two different algebraic methods to simplify ${4}^{\frac{3}{2}}.$ Explain all your steps.
Explain why the expression ${\left(\mathrm{-16}\right)}^{\frac{3}{2}}$ cannot be evaluated.
Simplify Expressions with Square Roots
In the following exercises, simplify.
$\sqrt{64}$
$-\sqrt{25}$
$\sqrt{\mathrm{-9}}$
$\sqrt{64}+\sqrt{225}$
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
$\sqrt{28}$
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
$\sqrt{15}$
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.
$\sqrt{{q}^{2}}$
$\text{\u2212}\sqrt{121{a}^{2}}$
$\text{\u2212}\sqrt{100{q}^{2}}$
$\sqrt{4{a}^{2}{b}^{2}}$
Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
$\sqrt{300}$
$\sqrt{{x}^{13}}$
$\sqrt{16{m}^{4}}$
$\sqrt{288{m}^{21}}$
$\sqrt{48{r}^{5}{s}^{4}}$
$\frac{10-\sqrt{50}}{5}$
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.
$\sqrt{\frac{16}{25}}$
$\sqrt{\frac{{x}^{8}}{{x}^{4}}}$
$\sqrt{\frac{98{p}^{6}}{2{p}^{2}}}$
$\sqrt{\frac{65}{121}}$
$\sqrt{\frac{64{x}^{4}}{25{x}^{2}}}$
$\sqrt{\frac{36{r}^{10}}{16{r}^{5}}}$
$\frac{3{r}^{2}\sqrt{r}}{2}$
$\sqrt{\frac{48{p}^{3}{q}^{5}}{27pq}}$
$\sqrt{\frac{12{r}^{5}{s}^{7}}{75{r}^{2}s}}$
$\frac{2r{s}^{3}\sqrt{r}}{5}$
Add and Subtract Like Square Roots
In the following exercises, simplify.
$3\sqrt{2}+\sqrt{2}$
$4\sqrt{y}+4\sqrt{y}$
$\mathrm{-3}\sqrt{7}+2\sqrt{7}-\sqrt{7}$
$3\sqrt{5xy}-\sqrt{5xy}+3\sqrt{5xy}$
$2\sqrt{3rs}+\sqrt{3rs}-5\sqrt{rs}$
$3\sqrt{3rs}-5\sqrt{rs}$
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.
$\sqrt{32}+3\sqrt{2}$
$\sqrt{72}+\sqrt{50}$
$3\sqrt{32}+\sqrt{98}$
$\sqrt{50{y}^{5}}-\sqrt{72{y}^{5}}$
$6\sqrt{18{n}^{4}}-3\sqrt{8{n}^{4}}+{n}^{2}\sqrt{50}$
$17{n}^{2}\sqrt{2}$
Multiply Square Roots
In the following exercises, simplify.
$\sqrt{2}\xb7\sqrt{20}$
$\sqrt{2{m}^{2}}\xb7\sqrt{20{m}^{4}}$
$\left(6\sqrt{2y}\right)\left(3\sqrt{50{y}^{3}}\right)$
$180{y}^{2}$
$\left(6\sqrt{3{v}^{4}}\right)\left(5\sqrt{30v}\right)$
${\left(\text{\u2212}\sqrt{10}\right)}^{2}$
$\left(\mathrm{-3}\sqrt{3}\right)\left(5\sqrt{18}\right)$
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.
$\sqrt{3}\left(4+\sqrt{12}\right)$
$\left(5+\sqrt{2}\right)\left(3-\sqrt{2}\right)$
$13-2\sqrt{2}$
$\left(5-3\sqrt{7}\right)\left(1-2\sqrt{7}\right)$
$\left(1-3\sqrt{x}\right)\left(5+2\sqrt{x}\right)$
$5-13\sqrt{x}-6x$
$\left(3+4\sqrt{y}\right)\left(10-\sqrt{y}\right)$
${\left(2-6\sqrt{5}\right)}^{2}$
$\left(3+2\sqrt{7}\right)\left(3-2\sqrt{7}\right)$
$\mathrm{-19}$
$\left(6-\sqrt{11}\right)\left(6+\sqrt{11}\right)$
Divide Square Roots
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