8.3 Simplify 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: ⓐ $b^{\frac{1}{6}}$ ⓑ $z^{\frac{1}{5}}$ ⓒ $p^{\frac{1}{4}} .$

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

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: ⓐ $\sqrt{x^{5}}$ ⓑ $\left(\sqrt[4]{3 y}\right)^{3}$ ⓒ $\sqrt{\left(\frac{2 m}{3 n}\right)^{5}} .$

Write with a rational exponent: ⓐ $\sqrt[5]{a^{2}}$ ⓑ $\left(\sqrt[3]{5 a b}\right)^{5}$ ⓒ $\sqrt{\left(\frac{7 x y}{z}\right)^{3}} .$

Simplify: ⓐ $125^{\frac{2}{3}}$ ⓑ $16^{- \frac{3}{2}}$ ⓒ $32^{- \frac{2}{5}} .$

Simplify: ⓐ $27^{\frac{2}{3}}$ ⓑ $81^{- \frac{3}{2}}$ ⓒ $16^{- \frac{3}{4}} .$

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: ⓐ $x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}$ ⓑ $\left(z^{9}\right)^{\frac{2}{3}}$ ⓒ $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}} .$

Simplify: ⓐ $x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}$ ⓑ $\left(x^{6}\right)^{\frac{4}{3}}$ ⓒ $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}} .$

Simplify: ⓐ $y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}$ ⓑ $\left(m^{9}\right)^{\frac{2}{9}}$ ⓒ $\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}} .$

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

Simplify: ⓐ $\left(32 x^{\frac{1}{3}}\right)^{\frac{3}{5}}$ ⓑ $\left(x^{\frac{3}{4}} y^{\frac{1}{2}}\right)^{\frac{2}{3}} .$

Simplify: ⓐ $\left(81 n^{\frac{2}{5}}\right)^{\frac{3}{2}}$ ⓑ $\left(a^{\frac{3}{2}} b^{\frac{1}{2}}\right)^{\frac{4}{3}} .$

Simplify: ⓐ $\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}$ ⓑ $\left(\frac{16 x^{\frac{4}{3}} y^{- \frac{5}{6}}}{x^{- \frac{2}{3}} y^{\frac{1}{6}}}\right)^{\frac{1}{2}} .$

Simplify: ⓐ $\frac{m^{\frac{2}{3}} \cdot m^{- \frac{1}{3}}}{m^{- \frac{5}{3}}}$ ⓑ $\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{- \frac{1}{6}}}\right)^{\frac{1}{2}} .$

Simplify: ⓐ $\frac{u^{\frac{4}{5}} \cdot u^{- \frac{2}{5}}}{u^{- \frac{13}{5}}}$ ⓑ $\left(\frac{27 x^{\frac{4}{5}} y^{\frac{1}{6}}}{x^{\frac{1}{5}} y^{- \frac{5}{6}}}\right)^{\frac{1}{3}} .$

Write as a radical expression.

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

Write as a radical expression.

ⓐ $u^{\frac{1}{5}}$ ⓑ $v^{\frac{1}{9}}$ ⓒ $w^{\frac{1}{20}}$

Write with a rational exponent.

ⓐ $\sqrt[7]{x}$ ⓑ $\sqrt[9]{y}$ ⓒ $\sqrt[5]{f}$

Write with a rational exponent.

ⓐ $\sqrt[3]{7 c}$ ⓑ $\sqrt[7]{12 d}$ ⓒ $2 \sqrt[4]{6 b}$

Write with a rational exponent.

ⓐ $\sqrt{21 p}$ ⓑ $\sqrt[4]{8 q}$ ⓒ $4 \sqrt[6]{36 r}$

Simplify.

ⓐ $81^{\frac{1}{2}}$ ⓑ $125^{\frac{1}{3}}$ ⓒ $64^{\frac{1}{2}}$

Simplify.

ⓐ $16^{\frac{1}{4}}$ ⓑ $16^{\frac{1}{2}}$ ⓒ $625^{\frac{1}{4}}$

Simplify.

ⓐ $\left(−216\right)^{\frac{1}{3}}$ ⓑ $− 216^{\frac{1}{3}}$ ⓒ $\left(216\right)^{- \frac{1}{3}}$

Simplify.

ⓐ $\left(−81\right)^{\frac{1}{4}}$ ⓑ $− 81^{\frac{1}{4}}$ ⓒ $\left(81\right)^{- \frac{1}{4}}$

Simplify.

ⓐ $\left(−36\right)^{\frac{1}{2}}$ ⓑ $− 36^{\frac{1}{2}}$ ⓒ $\left(36\right)^{- \frac{1}{2}}$

Simplify.

ⓐ $\left(−100\right)^{\frac{1}{2}}$ ⓑ $− 100^{\frac{1}{2}}$ ⓒ $\left(100\right)^{- \frac{1}{2}}$

Write with a rational exponent.

ⓐ $\sqrt{m^{5}}$ ⓑ $\left(\sqrt[3]{3 y}\right)^{7}$ ⓒ $\sqrt[5]{\left(\frac{4 x}{5 y}\right)^{3}}$

Write with a rational exponent.

ⓐ $\sqrt[5]{u^{2}}$ ⓑ $\left(\sqrt[3]{6 x}\right)^{5}$ ⓒ $\sqrt[4]{\left(\frac{18 a}{5 b}\right)^{7}}$

Simplify.

ⓐ $64^{\frac{5}{2}}$ ⓑ $81^{\frac{−3}{2}}$ ⓒ $\left(−27\right)^{\frac{2}{3}}$

Simplify.

ⓐ $32^{\frac{2}{5}}$ ⓑ $27^{- \frac{2}{3}}$ ⓒ $\left(−25\right)^{\frac{1}{2}}$

Simplify.

ⓐ $− 9^{\frac{3}{2}}$ ⓑ $− 9^{- \frac{3}{2}}$ ⓒ $\left(−9\right)^{\frac{3}{2}}$

Simplify. Assume all variables are positive.

ⓐ $c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}$ ⓑ $\left(p^{12}\right)^{\frac{3}{4}}$ ⓒ $\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$

Simplify. Assume all variables are positive.

ⓐ $y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}$ ⓑ $\left(x^{12}\right)^{\frac{2}{3}}$ ⓒ $\frac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}$

Simplify. Assume all variables are positive.

ⓐ $\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}$ ⓑ $\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}$

Simplify. Assume all variables are positive.

ⓐ $\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}$ ⓑ $\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}$

Simplify. Assume all variables are positive.

ⓐ $\frac{r^{\frac{5}{2}} \cdot r^{- \frac{1}{2}}}{r^{- \frac{3}{2}}}$ ⓑ $\left(\frac{36 s^{\frac{1}{5}} t^{- \frac{3}{2}}}{s^{- \frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}$

Simplify. Assume all variables are positive.

ⓐ $\frac{c^{\frac{5}{3}} \cdot c^{- \frac{1}{3}}}{c^{- \frac{2}{3}}}$ ⓑ $\left(\frac{8 x^{\frac{5}{3}} y^{- \frac{1}{2}}}{27 x^{- \frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}$

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