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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: ⓐ ${\left({m}^{3}{n}^{9}\right)}^{\frac{1}{3}}$ ⓑ ${\left({p}^{4}{q}^{8}\right)}^{\frac{1}{4}}$ .
We will use both the Product and Quotient Properties in the next example.
Simplify: ⓐ $\frac{{x}^{\frac{3}{4}}\xb7{x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}}$ ⓑ $\frac{{y}^{\frac{4}{3}}\xb7y}{{y}^{-\frac{2}{3}}}$ .
Simplify: ⓐ $\frac{{m}^{\frac{2}{3}}\xb7{m}^{-\frac{1}{3}}}{{m}^{-\frac{5}{3}}}$ ⓑ $\frac{{n}^{\frac{1}{6}}\xb7n}{{n}^{-\frac{11}{6}}}$ .
ⓐ ${m}^{2}$ ⓑ ${n}^{3}$
Simplify: ⓐ $\frac{{u}^{\frac{4}{5}}\xb7{u}^{-\frac{2}{5}}}{{u}^{-\frac{13}{5}}}$ ⓑ $\frac{{v}^{\frac{1}{2}}\xb7v}{{v}^{-\frac{7}{2}}}$ .
ⓐ ${u}^{3}$ ⓑ ${v}^{5}$
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
ⓐ
${\left(\mathrm{-216}\right)}^{\frac{1}{3}}$
ⓑ
$\text{\u2212}{216}^{\frac{1}{3}}$
ⓒ
${\left(216\right)}^{-\frac{1}{3}}$
ⓐ
${\left(\mathrm{-243}\right)}^{\frac{1}{5}}$
ⓑ
$\text{\u2212}{243}^{\frac{1}{5}}$
ⓒ
${\left(243\right)}^{-\frac{1}{5}}$
ⓐ $\mathrm{-3}$ ⓑ $\mathrm{-3}$ ⓒ $\frac{1}{3}$
ⓐ
${\left(\mathrm{-1}\right)}^{\frac{1}{3}}$
ⓑ
${\mathrm{-1}}^{\frac{1}{3}}$
ⓒ
${\left(1\right)}^{-\frac{1}{3}}$
ⓐ
${\left(\mathrm{-1000}\right)}^{\frac{1}{3}}$
ⓑ
$\text{\u2212}{1000}^{\frac{1}{3}}$
ⓒ
${\left(1000\right)}^{-\frac{1}{3}}$
ⓐ $\mathrm{-10}$ ⓑ $\mathrm{-10}$ ⓒ $\frac{1}{10}$
ⓐ
${\left(\mathrm{-81}\right)}^{\frac{1}{4}}$
ⓑ
$\text{\u2212}{81}^{\frac{1}{4}}$
ⓒ
${\left(81\right)}^{-\frac{1}{4}}$
ⓐ
${\left(\mathrm{-49}\right)}^{\frac{1}{2}}$
ⓑ
$\text{\u2212}{49}^{\frac{1}{2}}$
ⓒ
${\left(49\right)}^{-\frac{1}{2}}$
ⓐ not a real number ⓑ $\mathrm{-7}$ ⓒ $\frac{1}{7}$
ⓐ
${\left(\mathrm{-36}\right)}^{\frac{1}{2}}$
ⓑ
$-{36}^{\frac{1}{2}}$
ⓒ
${\left(36\right)}^{-\frac{1}{2}}$
ⓐ
${\left(\mathrm{-1}\right)}^{\frac{1}{4}}$
ⓑ
${\left(1\right)}^{-\frac{1}{4}}$
ⓒ
$\text{\u2212}{1}^{\frac{1}{4}}$
ⓐ not a real number ⓑ $1$ ⓒ $\mathrm{-1}$
ⓐ
${\left(\mathrm{-100}\right)}^{\frac{1}{2}}$
ⓑ
$\text{\u2212}{100}^{\frac{1}{2}}$
ⓒ
${\left(100\right)}^{-\frac{1}{2}}$
ⓐ
${\left(\mathrm{-32}\right)}^{\frac{1}{5}}$
ⓑ
${\left(243\right)}^{-\frac{1}{5}}$
ⓒ
$\text{\u2212}{125}^{\frac{1}{3}}$
ⓐ
$\mathrm{-2}$
ⓑ
$\frac{1}{3}$
ⓒ
$\mathrm{-5}$
Simplify Expressions with ${a}^{{\scriptscriptstyle \frac{m}{n}}}$
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.
ⓐ
${16}^{\frac{3}{2}}$
ⓑ
${8}^{\frac{2}{3}}$
ⓒ
${\mathrm{10,000}}^{\frac{3}{4}}$
ⓐ
${1000}^{\frac{2}{3}}$
ⓑ
${25}^{\frac{3}{2}}$
ⓒ
${32}^{\frac{3}{5}}$
ⓐ 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 ⓑ $\frac{1}{729}$ ⓒ $\frac{1}{81}$
ⓐ
${25}^{\frac{3}{2}}$
ⓑ
${9}^{-\frac{3}{2}}$
ⓒ
${\left(\mathrm{-64}\right)}^{\frac{2}{3}}$
ⓐ
${100}^{\frac{3}{2}}$
ⓑ
${49}^{-\frac{5}{2}}$
ⓒ
${\left(\mathrm{-100}\right)}^{\frac{3}{2}}$
ⓐ 1000 ⓑ $\frac{1}{\mathrm{16,807}}$ ⓒ not a real number
ⓐ
$\text{\u2212}{9}^{\frac{3}{2}}$
ⓑ
$\text{\u2212}{9}^{-\frac{3}{2}}$
ⓒ
${\left(\mathrm{-9}\right)}^{\frac{3}{2}}$
ⓐ
$\text{\u2212}{64}^{\frac{3}{2}}$
ⓑ
$\text{\u2212}{64}^{-\frac{3}{2}}$
ⓒ
${\left(\mathrm{-64}\right)}^{\frac{3}{2}}$
ⓐ
$\mathrm{-512}$
ⓑ
$-\frac{1}{512}$
ⓒ not a real number
ⓐ
$\text{\u2212}{100}^{\frac{3}{2}}$
ⓑ
$\text{\u2212}{100}^{-\frac{3}{2}}$
ⓒ
${\left(\mathrm{-100}\right)}^{\frac{3}{2}}$
ⓐ
$\text{\u2212}{49}^{\frac{3}{2}}$
ⓑ
$\text{\u2212}{49}^{-\frac{3}{2}}$
ⓒ
${\left(\mathrm{-49}\right)}^{\frac{3}{2}}$
ⓐ $\mathrm{-343}$ ⓑ $-\frac{1}{343}$ ⓒ not a real number
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.
ⓐ
${4}^{\frac{5}{8}}\xb7{4}^{\frac{11}{8}}$
ⓑ
${m}^{\frac{7}{12}}\xb7{m}^{\frac{17}{12}}$
ⓒ
${p}^{\frac{3}{7}}\xb7{p}^{\frac{18}{7}}$
ⓐ
${6}^{\frac{5}{2}}\xb7{6}^{\frac{1}{2}}$
ⓑ
${n}^{\frac{2}{10}}\xb7{n}^{\frac{8}{10}}$
ⓒ
${q}^{\frac{2}{5}}\xb7{q}^{\frac{13}{5}}$
ⓐ 216 ⓑ $n$ ⓒ ${q}^{3}$
ⓐ
${5}^{\frac{1}{2}}\xb7{5}^{\frac{7}{2}}$
ⓑ
${c}^{\frac{3}{4}}\xb7{c}^{\frac{9}{4}}$
ⓒ
${d}^{\frac{3}{5}}\xb7{d}^{\frac{2}{5}}$
ⓐ
${10}^{\frac{1}{3}}\xb7{10}^{\frac{5}{3}}$
ⓑ
${x}^{\frac{5}{6}}\xb7{x}^{\frac{7}{6}}$
ⓒ
${y}^{\frac{11}{8}}\xb7{y}^{\frac{21}{8}}$
ⓐ 100 ⓑ ${x}^{2}$ ⓒ ${y}^{4}$
ⓐ
${\left({m}^{6}\right)}^{\frac{5}{2}}$
ⓑ
${\left({n}^{9}\right)}^{\frac{4}{3}}$
ⓒ
${\left({p}^{12}\right)}^{\frac{3}{4}}$
ⓐ
${\left({a}^{12}\right)}^{\frac{1}{6}}$
ⓑ
${\left({b}^{15}\right)}^{\frac{3}{5}}$
ⓒ
${\left({c}^{11}\right)}^{\frac{1}{11}}$
ⓐ
${a}^{2}$
ⓑ
${b}^{9}$
ⓒ
$c$
ⓐ
${\left({x}^{12}\right)}^{\frac{2}{3}}$
ⓑ
${\left({y}^{20}\right)}^{\frac{2}{5}}$
ⓒ
${\left({z}^{16}\right)}^{\frac{1}{16}}$
ⓐ
${\left({h}^{6}\right)}^{\frac{4}{3}}$
ⓑ
${\left({k}^{12}\right)}^{\frac{3}{4}}$
ⓒ
${\left({j}^{10}\right)}^{\frac{7}{5}}$
ⓐ ${h}^{8}$ ⓑ ${k}^{9}$ ⓒ ${j}^{14}$
ⓐ
$\frac{{x}^{\frac{7}{2}}}{{x}^{\frac{5}{2}}}$
ⓑ
$\frac{{y}^{\frac{5}{2}}}{{y}^{\frac{1}{2}}}$
ⓒ
$\frac{{r}^{\frac{4}{5}}}{{r}^{\frac{9}{5}}}$
ⓐ
$\frac{{s}^{\frac{11}{5}}}{{s}^{\frac{6}{5}}}$
ⓑ
$\frac{{z}^{\frac{7}{3}}}{{z}^{\frac{1}{3}}}$
ⓒ
$\frac{{w}^{\frac{2}{7}}}{{w}^{\frac{9}{7}}}$
ⓐ $s$ ⓑ ${z}^{2}$ ⓒ $\frac{1}{w}$
ⓐ
$\frac{{t}^{\frac{12}{5}}}{{t}^{\frac{7}{5}}}$
ⓑ
$\frac{{x}^{\frac{3}{2}}}{{x}^{\frac{1}{2}}}$
ⓒ
$\frac{{m}^{\frac{13}{8}}}{{m}^{\frac{5}{8}}}$
ⓐ
$\frac{{u}^{\frac{13}{9}}}{{u}^{\frac{4}{9}}}$
ⓑ
$\frac{{r}^{\frac{15}{7}}}{{r}^{\frac{8}{7}}}$
ⓒ
$\frac{{n}^{\frac{3}{5}}}{{n}^{\frac{8}{5}}}$
ⓐ $u$ ⓑ $r$ ⓒ $\frac{1}{n}$
ⓐ
${\left(9{p}^{\frac{2}{3}}\right)}^{\frac{5}{2}}$
ⓑ
${\left(27{q}^{\frac{3}{2}}\right)}^{\frac{4}{3}}$
ⓐ
${\left(81{r}^{\frac{4}{5}}\right)}^{\frac{1}{4}}$
ⓑ
${\left(64{s}^{\frac{3}{7}}\right)}^{\frac{1}{6}}$
ⓐ $3{r}^{\frac{1}{5}}$ ⓑ $2{s}^{\frac{1}{14}}$
ⓐ
${\left(16{u}^{\frac{1}{3}}\right)}^{\frac{3}{4}}$
ⓑ
${\left(100{v}^{\frac{2}{5}}\right)}^{\frac{3}{2}}$
ⓐ
${\left(27{m}^{\frac{3}{4}}\right)}^{\frac{2}{3}}$
ⓑ
${\left(625{n}^{\frac{8}{3}}\right)}^{\frac{3}{4}}$
ⓐ $9{m}^{\frac{1}{2}}$ ⓑ $125{n}^{2}$
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