9.8 Rational Exponents

Before you get started, take this readiness quiz.

Add: $\frac{7}{15} + \frac{5}{12}$.
If you missed this problem, review the reference.

Simplify: $\left(4 x^{2} y^{5}\right)^{3}$.
If you missed this problem, review the reference.

Simplify: $5^{−3}$.
If you missed this problem, review the reference.

Write as a radical expression: ⓐ $x^{\frac{1}{2}}$ ⓑ $y^{\frac{1}{3}}$ ⓒ $z^{\frac{1}{4}}$.

Write as a radical expression: ⓐ $t^{\frac{1}{2}}$ ⓑ $m^{\frac{1}{3}}$ ⓒ $r^{\frac{1}{4}}$.

Write as a radial expression: ⓐ $b^{\frac{1}{2}}$ ⓑ $z^{\frac{1}{3}}$ ⓒ $p^{\frac{1}{4}}$.

Write with a rational exponent: ⓐ $\sqrt{x}$ ⓑ $\sqrt[3]{y}$ ⓒ $\sqrt[4]{z}$.

Write with a rational exponent: ⓐ $\sqrt{s}$ ⓑ $\sqrt[3]{x}$ ⓒ $\sqrt[4]{b}$.

Write with a rational exponent: ⓐ $\sqrt{v}$ ⓑ $\sqrt[3]{p}$ ⓒ $\sqrt[4]{p}$.

Write with a rational exponent: ⓐ $\sqrt{5 y}$ ⓑ $\sqrt[3]{4 x}$ ⓒ $3 \sqrt[4]{5 z}$.

Write with a rational exponent: ⓐ $\sqrt{10 m}$ ⓑ $\sqrt[5]{3 n}$ ⓒ $3 \sqrt[4]{6 y}$.

Write with a rational exponent: ⓐ $\sqrt[7]{3 k}$ ⓑ $\sqrt[4]{5 j}$ ⓒ $8 \sqrt[3]{2 a}$.

Simplify: ⓐ $25^{\frac{1}{2}}$ ⓑ $64^{\frac{1}{3}}$ ⓒ $256^{\frac{1}{4}}$.

Simplify: ⓐ $36^{\frac{1}{2}}$ ⓑ $8^{\frac{1}{3}}$ ⓒ $16^{\frac{1}{4}}$.

Simplify: ⓐ $100^{\frac{1}{2}}$ ⓑ $27^{\frac{1}{3}}$ ⓒ $81^{\frac{1}{4}}$.

Simplify: ⓐ $\left(−64\right)^{\frac{1}{3}}$ ⓑ $− 64^{\frac{1}{3}}$ ⓒ $\left(64\right)^{- \frac{1}{3}}$.

Simplify: ⓐ $\left(−125\right)^{\frac{1}{3}}$ ⓑ $− 125^{\frac{1}{3}}$ ⓒ $\left(125\right)^{- \frac{1}{3}}$.

Simplify: ⓐ $\left(−32\right)^{\frac{1}{5}}$ ⓑ $− 32^{\frac{1}{5}}$ ⓒ $\left(32\right)^{- \frac{1}{5}}$.

Simplify: ⓐ $\left(−16\right)^{\frac{1}{4}}$ ⓑ $− 16^{\frac{1}{4}}$ ⓒ $\left(16\right)^{- \frac{1}{4}}$.

Simplify: ⓐ $\left(−64\right)^{\frac{1}{2}}$ ⓑ $− 64^{\frac{1}{2}}$ ⓒ $\left(64\right)^{- \frac{1}{2}}$.

Simplify: ⓐ $\left(−256\right)^{\frac{1}{4}}$ ⓑ $− 256^{\frac{1}{4}}$ ⓒ $\left(256\right)^{- \frac{1}{4}}$.

Write with a rational exponent: ⓐ $\sqrt{x^{5}}$ ⓑ $\sqrt[4]{z^{3}}$ ⓒ $\sqrt[5]{y^{2}}$.

Write with a rational exponent: ⓐ $\sqrt[5]{a^{2}}$ ⓑ $\sqrt[3]{b^{7}}$ ⓒ $\sqrt[4]{m^{5}}$.

Simplify: ⓐ $9^{\frac{3}{2}}$ ⓑ $125^{\frac{2}{3}}$ ⓒ $81^{\frac{3}{4}}$.

Simplify: ⓐ $4^{\frac{3}{2}}$ ⓑ $27^{\frac{2}{3}}$ ⓒ $625^{\frac{3}{4}}$.

Simplify: ⓐ $8^{\frac{5}{3}}$ ⓑ $81^{\frac{3}{2}}$ ⓒ $16^{\frac{3}{4}}$.

Simplify: ⓐ $16^{- \frac{3}{2}}$ ⓑ $32^{- \frac{2}{5}}$ ⓒ $4^{- \frac{5}{2}}$.

Simplify: ⓐ $8^{- \frac{5}{3}}$ ⓑ $81^{- \frac{3}{2}}$ ⓒ $16^{- \frac{3}{4}}$.

Simplify: ⓐ $4^{- \frac{3}{2}}$ ⓑ $27^{- \frac{2}{3}}$ ⓒ $625^{- \frac{3}{4}}$.

Simplify: ⓐ $− 25^{\frac{3}{2}}$ ⓑ $− 25^{- \frac{3}{2}}$ ⓒ $\left(−25\right)^{\frac{3}{2}}$.

Simplify: ⓐ $−16^{\frac{3}{2}}$ ⓑ $−16^{- \frac{3}{2}}$ ⓒ $\left(−16\right)^{- \frac{3}{2}}$.

Simplify: ⓐ $−81^{\frac{3}{2}}$ ⓑ $−81^{- \frac{3}{2}}$ ⓒ $\left(−81\right)^{- \frac{3}{2}}$.

Simplify: ⓐ $2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}}$ ⓑ $x^{\frac{2}{3}} \cdot x^{\frac{4}{3}}$ ⓒ $z^{\frac{3}{4}} \cdot z^{\frac{5}{4}}$.

Simplify: ⓐ $3^{\frac{2}{3}} \cdot 3^{\frac{4}{3}}$ ⓑ $y^{\frac{1}{3}} \cdot y^{\frac{8}{3}}$ ⓒ $m^{\frac{1}{4}} \cdot m^{\frac{3}{4}}$.

Simplify: ⓐ $5^{\frac{3}{5}} \cdot 5^{\frac{7}{5}}$ ⓑ $z^{\frac{1}{8}} \cdot z^{\frac{7}{8}}$ ⓒ $n^{\frac{2}{7}} \cdot n^{\frac{5}{7}}$.

Simplify: ⓐ $\left(x^{4}\right)^{\frac{1}{2}}$ ⓑ $\left(y^{6}\right)^{\frac{1}{3}}$ ⓒ $\left(z^{9}\right)^{\frac{2}{3}}$.

Simplify: ⓐ $\left(p^{10}\right)^{\frac{1}{5}}$ ⓑ $\left(q^{8}\right)^{\frac{3}{4}}$ ⓒ $\left(x^{6}\right)^{\frac{4}{3}}$.

Simplify: ⓐ $\left(r^{6}\right)^{\frac{5}{3}}$ ⓑ $\left(s^{12}\right)^{\frac{3}{4}}$ ⓒ $\left(m^{9}\right)^{\frac{2}{9}}$.

Simplify: ⓐ $\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$ ⓑ $\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$ ⓒ $\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$.

Simplify: ⓐ $\frac{u^{\frac{5}{4}}}{u^{\frac{1}{4}}}$ ⓑ $\frac{v^{\frac{3}{5}}}{v^{\frac{2}{5}}}$ ⓒ $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$.

Simplify: ⓐ $\frac{c^{\frac{12}{5}}}{c^{\frac{2}{5}}}$ ⓑ $\frac{m^{\frac{5}{4}}}{m^{\frac{9}{4}}}$ ⓒ $\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}$.

Simplify: ⓐ $\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}$ ⓑ $\left(8 v^{\frac{1}{4}}\right)^{\frac{2}{3}}$.

Simplify: ⓐ $\left(32 x^{\frac{1}{3}}\right)^{\frac{3}{5}}$ ⓑ $\left(64 y^{\frac{2}{3}}\right)^{\frac{1}{3}}$.

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

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

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

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

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

Write as a radical expression.

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

Write as a radical expression.

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

Write with a rational exponent.

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

Simplify.

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

Simplify.

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

Simplify.

ⓐ $\left(−243\right)^{\frac{1}{5}}$ ⓑ $− 243^{\frac{1}{5}}$ ⓒ $\left(243\right)^{- \frac{1}{5}}$

Simplify.

ⓐ $\left(−1000\right)^{\frac{1}{3}}$ ⓑ $− 1000^{\frac{1}{3}}$ ⓒ $\left(1000\right)^{- \frac{1}{3}}$

Simplify.

ⓐ $\left(−49\right)^{\frac{1}{2}}$ ⓑ $− 49^{\frac{1}{2}}$ ⓒ $\left(49\right)^{- \frac{1}{2}}$

Simplify.

ⓐ $\left(−1\right)^{\frac{1}{4}}$ ⓑ $\left(1\right)^{- \frac{1}{4}}$ ⓒ $− 1^{\frac{1}{4}}$

Simplify.

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

Write with a rational exponent.

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

Write with a rational exponent.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

ⓐ $− 64^{\frac{3}{2}}$ ⓑ $− 64^{- \frac{3}{2}}$ ⓒ $\left(−64\right)^{\frac{3}{2}}$

Simplify.

ⓐ $− 49^{\frac{3}{2}}$ ⓑ $− 49^{- \frac{3}{2}}$ ⓒ $\left(−49\right)^{\frac{3}{2}}$

Simplify.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$ ⓑ $n^{\frac{2}{10}} \cdot n^{\frac{8}{10}}$ ⓒ $q^{\frac{2}{5}} \cdot q^{\frac{13}{5}}$

Simplify.

ⓐ $10^{\frac{1}{3}} \cdot 10^{\frac{5}{3}}$ ⓑ $x^{\frac{5}{6}} \cdot x^{\frac{7}{6}}$ ⓒ $y^{\frac{11}{8}} \cdot y^{\frac{21}{8}}$

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

ⓐ $\frac{m^{\frac{7}{4}} \cdot m^{- \frac{5}{4}}}{m^{- \frac{2}{4}}}$ ⓑ $\frac{n^{\frac{3}{7}} \cdot n}{n^{- \frac{4}{7}}}$

Simplify.

$n^{\frac{2}{6}} \cdot n^{\frac{4}{6}}$

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

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.

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.

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

Simplify.

$\sqrt{144}$

Simplify.

$- \sqrt{81}$

Simplify.

$\sqrt{−36}$

Simplify.

$\sqrt{64 + 225}$

Estimate each square root between two consecutive whole numbers.

$\sqrt{155}$

Approximate each square root and round to two decimal places.

$\sqrt{57}$

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

$\sqrt{98}$

Simplify.

$\sqrt{y^{19}}$

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

$2 \sqrt{3 r s} + \sqrt{3 r s} - 5 \sqrt{r s}$

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

$2 \sqrt{2} \cdot 6 \sqrt{14}$

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

$\frac{\sqrt{75}}{10}$

Simplify.

$\frac{\sqrt{48}}{\sqrt{27}}$

Simplify.

$\frac{\sqrt{20 y^{5}}}{\sqrt{2 y}}$

Rationalize the denominator.

$\frac{10}{\sqrt{15}}$

Rationalize the denominator.

$\frac{5}{3 \sqrt{5}}$

Rationalize the denominator.

$\sqrt{\frac{3}{28}}$

Rationalize the denominator.

$\frac{4}{4 + \sqrt{27}}$

Rationalize the denominator.

$\frac{4}{2 - \sqrt{5}}$

Rationalize the denominator.

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

Solve the equation.

$\sqrt{7 z + 1} = 6$

Solve the equation.

$\sqrt{6 m + 4} - 5 = 0$

Solve the equation.

$\sqrt{u - 4} + 4 = u$

Solve the equation.

$\sqrt{r - 4} - r = −10$

Solve the equation.

$2 \sqrt{2 x - 7} - 4 = 8$

Solve the equation.

$\sqrt{a} + 3 = \sqrt{a + 9}$

Solve the equation.

$\sqrt{u} + 2 = \sqrt{u + 5}$

Solve the equation.

$\sqrt{y + 5} + 1 = \sqrt{2 y + 3}$

Solve. Round approximations to one decimal place.

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker.

Simplify.

ⓐ $\sqrt[6]{64}$
ⓑ $\sqrt[3]{64}$

Simplify.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ $\sqrt[8]{v^{8}}$

Simplify.

ⓐ $\sqrt[4]{16 x^{8}}$
ⓑ $\sqrt[6]{64 y^{12}}$

Simplify.

ⓐ $\sqrt[5]{64 c^{8}}$
ⓑ $\sqrt[4]{48 d^{7}}$

Simplify.

ⓐ $\sqrt[3]{−500}$
ⓑ $\sqrt[4]{−16}$

Simplify.

$\sqrt[3]{\frac{w^{12}}{w^{2}}}$

Simplify.

$\sqrt[3]{\frac{54 z^{9}}{2 z^{3}}}$

Simplify.

$4 \sqrt[5]{20} - 2 \sqrt[5]{20}$

Simplify.

$\sqrt[4]{1250} - \sqrt[4]{162}$

Simplify.

$\sqrt[5]{96 t^{8}} + \sqrt[5]{486 t^{4}}$

Write with a rational exponent.

$\sqrt[6]{v}$

Write with a rational exponent.

$\sqrt[6]{10 z}$

Simplify.

$32^{\frac{1}{5}}$

Simplify.

$\left(125\right)^{- \frac{1}{3}}$

Simplify.

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

Write with a rational exponent.

$\sqrt[5]{n^{8}}$

Simplify.

$64^{\frac{5}{2}}$

Simplify.

$81^{- \frac{5}{2}}$

Simplify.

$\left(x^{6}\right)^{\frac{4}{3}}$

Simplify.

$\left(16 s^{\frac{9}{4}}\right)^{\frac{1}{4}}$

Simplify.

$\frac{z^{\frac{2}{3}} \cdot z^{- \frac{1}{3}}}{z^{- \frac{5}{3}}}$

Simplify.

$\sqrt{169 m^{4} n^{2}}$

Simplify.

$3 \sqrt{13} + 5 \sqrt{2} + \sqrt{13}$

Simplify.

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

Simplify.

$\left(1 - 2 \sqrt{q}\right)^{2}$

Simplify.

ⓐ $\sqrt[4]{81 x^{12}}$
ⓑ $\sqrt[6]{64 y^{18}}$

Simplify.

$\sqrt{\frac{14 y^{3}}{7 y}}$