8.5 Divide Radical Expressions

Before you get started, take this readiness quiz.

Simplify: $\frac{30}{48} .$
If you missed this problem, review the reference.

Simplify: $x^{2} \cdot x^{4} .$
If you missed this problem, review the reference.

Multiply: $\left(7 + 3 x\right) \left(7 - 3 x\right) .$
If you missed this problem, review the reference.

Simplify: ⓐ $\frac{\sqrt{72 x^{3}}}{\sqrt{162 x}}$ ⓑ $\frac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}} .$

Simplify: ⓐ $\frac{\sqrt{50 s^{3}}}{\sqrt{128 s}}$ ⓑ $\frac{\sqrt[3]{56 a^{}}}{\sqrt[3]{7 a^{4}}} .$

Simplify: ⓐ $\frac{\sqrt{75 q^{5}}}{\sqrt{108 q}}$ ⓑ $\frac{\sqrt[3]{72 b^{2}}}{\sqrt[3]{9 b^{5}}} .$

Simplify: ⓐ $\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$ ⓑ $\frac{\sqrt[3]{−250 m^{} n^{−2}}}{\sqrt[3]{2 m^{−2} n^{4}}} .$

Simplify: ⓐ $\frac{\sqrt{162 x^{10} y^{2}}}{\sqrt{2 x^{6} y^{6}}}$ ⓑ $\frac{\sqrt[3]{−128 x^{2} y^{−1}}}{\sqrt[3]{2 x^{−1} y^{2}}} .$

Simplify: ⓐ $\frac{\sqrt{300 m^{3} n^{7}}}{\sqrt{3 m^{5} n}}$ ⓑ $\frac{\sqrt[3]{−81 p q^{−1}}}{\sqrt[3]{3 p^{−2} q^{5}}} .$

Simplify: $\frac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}} .$

Simplify: $\frac{\sqrt{64 x^{4} y^{5}}}{\sqrt{2 x y^{3}}} .$

Simplify: $\frac{\sqrt{96 a^{5} b^{4}}}{\sqrt{2 a^{3} b}} .$

Simplify: ⓐ $\frac{5}{\sqrt{3}}$ ⓑ $\sqrt{\frac{3}{32}}$ ⓒ $\frac{2}{\sqrt{2 x}} .$

Simplify: ⓐ $\frac{6}{\sqrt{5}}$ ⓑ $\sqrt{\frac{7}{18}}$ ⓒ $\frac{5}{\sqrt{5 x}} .$

Simplify: ⓐ $\frac{1}{\sqrt[3]{7}}$ ⓑ $\sqrt[3]{\frac{5}{12}}$ ⓒ $\frac{5}{\sqrt[3]{9 y}} .$

Simplify: ⓐ $\frac{1}{\sqrt[3]{2}}$ ⓑ $\sqrt[3]{\frac{3}{20}}$ ⓒ $\frac{2}{\sqrt[3]{25 n}} .$

Simplify: ⓐ $\frac{1}{\sqrt[4]{3}}$ ⓑ $\sqrt[4]{\frac{3}{64}}$ ⓒ $\frac{3}{\sqrt[4]{125 x}} .$

Simplify: ⓐ $\frac{1}{\sqrt[4]{5}}$ ⓑ $\sqrt[4]{\frac{7}{128}}$ ⓒ $\frac{4}{\sqrt[4]{4 x}}$

Simplify: $\frac{3}{1 - \sqrt{5}} .$

Simplify: $\frac{2}{4 - \sqrt{6}} .$

Simplify: $\frac{\sqrt{5}}{\sqrt{x} + \sqrt{2}} .$

Simplify: $\frac{\sqrt{10}}{\sqrt{y} - \sqrt{3}} .$

Simplify: $\frac{\sqrt{p} + \sqrt{2}}{\sqrt{p} - \sqrt{2}} .$

Simplify: $\frac{\sqrt{q} - \sqrt{10}}{\sqrt{q} + \sqrt{10}}$

Simplify.

ⓐ $\frac{\sqrt{128}}{\sqrt{72}}$ ⓑ $\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$

Simplify.

ⓐ $\frac{\sqrt{200 m^{5}}}{\sqrt{98 m}}$ ⓑ $\frac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}$

Simplify.

ⓐ $\frac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}$ ⓑ $\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}$

Simplify.

ⓐ $\frac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}$ ⓑ $\frac{\sqrt[3]{−16 a^{4} b^{−2}}}{\sqrt[3]{2 a^{−2} b^{}}}$

Simplify.

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$ ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

Simplify.

$\frac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}$

Simplify.

$\frac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{- 1} b^{3}}}$

Rationalize the denominator.

ⓐ $\frac{10}{\sqrt{6}}$ ⓑ $\sqrt{\frac{4}{27}}$ ⓒ $\frac{10}{\sqrt{5 x}}$

Rationalize the denominator.

ⓐ $\frac{6}{\sqrt{7}}$ ⓑ $\sqrt{\frac{8}{45}}$ ⓒ $\frac{12}{\sqrt{3 p}}$

Rationalize the denominator.

ⓐ $\frac{1}{\sqrt[3]{5}}$ ⓑ $\sqrt[3]{\frac{5}{24}}$ ⓒ $\frac{4}{\sqrt[3]{36 a}}$

Rationalize the denominator.

ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3 x^{2}}}$

Rationalize the denominator.

ⓐ $\frac{1}{\sqrt[4]{7}}$ ⓑ $\sqrt[4]{\frac{5}{32}}$ ⓒ $\frac{4}{\sqrt[4]{4 x^{2}}}$

Rationalize the denominator.

ⓐ $\frac{1}{\sqrt[4]{9}}$ ⓑ $\sqrt[4]{\frac{25}{128}}$ ⓒ $\frac{6}{\sqrt[4]{27 a}}$

Simplify.

$\frac{8}{1 - \sqrt{5}}$

Simplify.

$\frac{6}{3 - \sqrt{7}}$

Simplify.

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

Simplify.

$\frac{\sqrt{2}}{\sqrt{x} - \sqrt{6}}$

Simplify.

$\frac{\sqrt{r} + \sqrt{5}}{\sqrt{r} - \sqrt{5}}$

Simplify.

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.
ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.
ⓒ Why are the two methods of simplifying square roots different?

Explain why multiplying $\sqrt{2 x} - 3$ by its conjugate results in an expression with no radicals.