Simplify the Radical Expression fifth root of 32/x

$\sqrt[5]{\frac{32}{x}}$

Step 1

Rewrite $\sqrt[5]{\frac{32}{x}}$ as $\frac{\sqrt[5]{32}}{\sqrt[5]{x}}$ .

$\frac{\sqrt[5]{32}}{\sqrt[5]{x}}$

Step 2

Simplify the numerator.

Rewrite $32$ as $2^{5}$ .

$\frac{\sqrt[5]{2^{5}}}{\sqrt[5]{x}}$

Pull terms out from under the radical.

$\frac{2}{\sqrt[5]{x}}$

Step 3

Multiply $\frac{2}{\sqrt[5]{x}}$ by $\frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$ .

$\frac{2}{\sqrt[5]{x}} \cdot \frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt[5]{x}}$ by $\frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$ .

$\frac{2 {\sqrt[5]{x}}^{4}}{\sqrt[5]{x} {\sqrt[5]{x}}^{4}}$

Raise $\sqrt[5]{x}$ to the power of $1$ .

$\frac{2 {\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{1} {\sqrt[5]{x}}^{4}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 {\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{2 {\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{5}}$

Rewrite ${\sqrt[5]{x}}^{5}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\frac{2 {\sqrt[5]{x}}^{4}}{{(x^{\frac{1}{5}})}^{5}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 {\sqrt[5]{x}}^{4}}{x^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{2 {\sqrt[5]{x}}^{4}}{x^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{2 {\sqrt[5]{x}}^{4}}{x^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{2 {\sqrt[5]{x}}^{4}}{x^{1}}$

Simplify.

$\frac{2 {\sqrt[5]{x}}^{4}}{x}$

Step 5

Rewrite ${\sqrt[5]{x}}^{4}$ as ${(x^{4})}^{\frac{1}{5}}$ .

$\frac{2 \sqrt[5]{x^{4}}}{x}$