8.2 Simplify Radical Expressions

Before you get started, take this readiness quiz.

Simplify: $\frac{x^{9}}{x^{4}} .$
If you missed this problem, review the reference.

Simplify: $\frac{y^{3}}{y^{11}} .$
If you missed this problem, review the reference.

Simplify: $\left(n^{2}\right)^{6} .$
If you missed this problem, review the reference.

Simplify: $\sqrt{48} .$

Simplify: $\sqrt{45} .$

Simplify: ⓐ $\sqrt{500}$ ⓑ $\sqrt[3]{16}$ ⓒ $\sqrt[4]{243} .$

Simplify: ⓐ $\sqrt{288}$ ⓑ $\sqrt[3]{81}$ ⓒ $\sqrt[4]{64} .$

Simplify: ⓐ $\sqrt{432}$ ⓑ $\sqrt[3]{625}$ ⓒ $\sqrt[4]{729} .$

Simplify: ⓐ $\sqrt{x^{3}}$ ⓑ $\sqrt[3]{x^{4}}$ ⓒ $\sqrt[4]{x^{7}} .$

Simplify: ⓐ $\sqrt{b^{5}}$ ⓑ $\sqrt[4]{y^{6}}$ ⓒ $\sqrt[3]{z^{5}}$

Simplify: ⓐ $\sqrt{p^{9}}$ ⓑ $\sqrt[5]{y^{8}}$ ⓒ $\sqrt[6]{q^{13}}$

Simplify: ⓐ $\sqrt{72 n^{7}}$ ⓑ $\sqrt[3]{24 x^{7}}$ ⓒ $\sqrt[4]{80 y^{14}} .$

Simplify: ⓐ $\sqrt{32 y^{5}}$ ⓑ $\sqrt[3]{54 p^{10}}$ ⓒ $\sqrt[4]{64 q^{10}} .$

Simplify: ⓐ $\sqrt{75 a^{9}}$ ⓑ $\sqrt[3]{128 m^{11}}$ ⓒ $\sqrt[4]{162 n^{7}} .$

Simplify: ⓐ $\sqrt{63 u^{3} v^{5}}$ ⓑ $\sqrt[3]{40 x^{4} y^{5}}$ ⓒ $\sqrt[4]{48 x^{4} y^{7}} .$

Simplify: ⓐ $\sqrt{98 a^{7} b^{5}}$ ⓑ $\sqrt[3]{56 x^{5} y^{4}}$ ⓒ $\sqrt[4]{32 x^{5} y^{8}} .$

Simplify: ⓐ $\sqrt{180 m^{9} n^{11}}$ ⓑ $\sqrt[3]{72 x^{6} y^{5}}$ ⓒ $\sqrt[4]{80 x^{7} y^{4}} .$

Simplify: ⓐ $\sqrt[3]{−27}$ ⓑ $\sqrt[4]{−16} .$

Simplify: ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−81} .$

Simplify: ⓐ $\sqrt[3]{−625}$ ⓑ $\sqrt[4]{−324} .$

Simplify: ⓐ $3 + \sqrt{32}$ ⓑ $\frac{4 - \sqrt{48}}{2} .$

Simplify: ⓐ $5 + \sqrt{75}$ ⓑ $\frac{10 - \sqrt{75}}{5}$

Simplify: ⓐ $2 + \sqrt{98}$ ⓑ $\frac{6 - \sqrt{45}}{3}$

Simplify: ⓐ $\sqrt{\frac{45}{80}}$ ⓑ $\sqrt[3]{\frac{16}{54}}$ ⓒ $\sqrt[4]{\frac{5}{80}} .$

Simplify: ⓐ $\sqrt{\frac{75}{48}}$ ⓑ $\sqrt[3]{\frac{54}{250}}$ ⓒ $\sqrt[4]{\frac{32}{162}} .$

Simplify: ⓐ $\sqrt{\frac{98}{162}}$ ⓑ $\sqrt[3]{\frac{24}{375}}$ ⓒ $\sqrt[4]{\frac{4}{324}} .$

Simplify: ⓐ $\sqrt{\frac{m^{6}}{m^{4}}}$ ⓑ $\sqrt[3]{\frac{a^{8}}{a^{5}}}$ ⓒ $\sqrt[4]{\frac{a^{10}}{a^{2}}} .$

Simplify: ⓐ $\sqrt{\frac{a^{8}}{a^{6}}}$ ⓑ $\sqrt[4]{\frac{x^{7}}{x^{3}}}$ ⓒ $\sqrt[4]{\frac{y^{17}}{y^{5}}} .$

Simplify: ⓐ $\sqrt{\frac{x^{14}}{x^{10}}}$ ⓑ $\sqrt[3]{\frac{m^{13}}{m^{7}}}$ ⓒ $\sqrt[5]{\frac{n^{12}}{n^{2}}} .$

Simplify: $\sqrt{\frac{24 p^{3}}{49}} .$

Simplify: $\sqrt{\frac{48 x^{5}}{100}} .$

Simplify: ⓐ $\sqrt{\frac{45 x^{5}}{y^{4}}}$ ⓑ $\sqrt[3]{\frac{24 x^{7}}{y^{3}}}$ ⓒ $\sqrt[4]{\frac{48 x^{10}}{y^{8}}} .$

Simplify: ⓐ $\sqrt{\frac{80 m^{3}}{n^{6}}}$ ⓑ $\sqrt[3]{\frac{108 c^{10}}{d^{6}}}$ ⓒ $\sqrt[4]{\frac{80 x^{10}}{y^{4}}} .$

Simplify: ⓐ $\sqrt{\frac{54 u^{7}}{v^{8}}}$ ⓑ $\sqrt[3]{\frac{40 r^{3}}{s^{6}}}$ ⓒ $\sqrt[4]{\frac{162 m^{14}}{n^{12}}} .$

Simplify: ⓐ $\sqrt{\frac{18 p^{5} q^{7}}{32 p q^{2}}}$ ⓑ $\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}} .$

Simplify: ⓐ $\sqrt{\frac{50 x^{5} y^{3}}{72 x^{4} y}}$ ⓑ $\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}} .$

Simplify: ⓐ $\sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}$ ⓑ $\sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}}$ ⓒ $\sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}} .$

Simplify: ⓐ $\frac{\sqrt{48 a^{7}}}{\sqrt{3 a}}$ ⓑ $\frac{\sqrt[3]{−108}}{\sqrt[3]{2}}$ ⓒ $\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}} .$

Simplify: ⓐ $\frac{\sqrt{98 z^{5}}}{\sqrt{2 z}}$ ⓑ $\frac{\sqrt[3]{−500}}{\sqrt[3]{2}}$ ⓒ $\frac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}} .$

Simplify: ⓐ $\frac{\sqrt{128 m^{9}}}{\sqrt{2 m}}$ ⓑ $\frac{\sqrt[3]{−192}}{\sqrt[3]{3}}$ ⓒ $\frac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}} .$

Use the Product Property to simplify radical expressions.

$\sqrt{27}$

Use the Product Property to simplify radical expressions.

$\sqrt{125}$

Use the Product Property to simplify radical expressions.

$\sqrt{147}$

Use the Product Property to simplify radical expressions.

$\sqrt{800}$

Use the Product Property to simplify radical expressions.

ⓐ $\sqrt[4]{32}$ ⓑ $\sqrt[5]{64}$

Use the Product Property to simplify radical expressions.

ⓐ $\sqrt[4]{64}$ ⓑ $\sqrt[3]{256}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{y^{11}}$ ⓑ $\sqrt[3]{r^{5}}$ ⓒ $\sqrt[4]{s^{10}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{n^{21}}$ ⓑ $\sqrt[3]{q^{8}}$ ⓒ $\sqrt[8]{n^{10}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{125 r^{13}}$ ⓑ $\sqrt[3]{108 x^{5}}$ ⓒ $\sqrt[4]{48 y^{6}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{242 m^{23}}$ ⓑ $\sqrt[4]{405 m^{10}}$ ⓒ $\sqrt[5]{160 n^{8}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{147 m^{7} n^{11}}$ ⓑ $\sqrt[3]{48 x^{6} y^{7}}$ ⓒ $\sqrt[4]{32 x^{5} y^{4}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt{192 q^{3} r^{7}}$ ⓑ $\sqrt[3]{54 m^{9} n^{10}}$ ⓒ $\sqrt[4]{81 a^{9} b^{8}}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt[3]{−864}$ ⓑ $\sqrt[4]{−256}$

Simplify using absolute value signs as needed.

ⓐ $\sqrt[5]{−32}$ ⓑ $\sqrt[8]{−1}$

Simplify using absolute value signs as needed.

ⓐ $5 + \sqrt{12}$ ⓑ $\frac{10 - \sqrt{24}}{2}$

Simplify using absolute value signs as needed.

ⓐ $1 + \sqrt{45}$ ⓑ $\frac{3 + \sqrt{90}}{3}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{45}{80}}$ ⓑ $\sqrt[3]{\frac{8}{27}}$ ⓒ $\sqrt[4]{\frac{1}{81}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{100}{36}}$ ⓑ $\sqrt[3]{\frac{81}{375}}$ ⓒ $\sqrt[4]{\frac{1}{256}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{x^{10}}{x^{6}}}$ ⓑ $\sqrt[3]{\frac{p^{11}}{p^{2}}}$ ⓒ $\sqrt[4]{\frac{q^{17}}{q^{13}}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{y^{4}}{y^{8}}}$ ⓑ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$ ⓒ $\sqrt[6]{\frac{v^{30}}{v^{12}}}$

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{96 x^{7}}{121}}$

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{300 m^{5}}{64}}$

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{98 r^{5}}{100}}$

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{28 q^{6}}{225}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{75 r^{9}}{s^{8}}}$ ⓑ $\sqrt[3]{\frac{54 a^{8}}{b^{3}}}$ ⓒ $\sqrt[4]{\frac{64 c^{5}}{d^{4}}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{28 p^{7}}{q^{2}}}$ ⓑ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$ ⓒ $\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$ ⓑ $\sqrt[3]{\frac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$

Use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$ ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$ ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

Use the Quotient Property to simplify square roots.

ⓐ $\frac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}$ ⓑ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓒ $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

Use the Quotient Property to simplify square roots.

ⓐ $\frac{\sqrt{50 m^{7}}}{\sqrt{2 m}}$ ⓑ $\sqrt[3]{\frac{1250}{2}}$ ⓒ $\sqrt[4]{\frac{486 y^{9}}{2 y^{3}}}$

Explain why $\sqrt{x^{4}} = x^{2} .$ Then explain why $\sqrt{x^{16}} = x^{8} .$

Explain how you know that $\sqrt[5]{x^{10}} = x^{2} .$