Simplify the Radical Expression fifth root of 64/r

$\sqrt[5]{\frac{64}{r}}$

Rewrite $\sqrt[5]{\frac{64}{r}}$ as $\frac{\sqrt[5]{64}}{\sqrt[5]{r}}$ .

$\frac{\sqrt[5]{64}}{\sqrt[5]{r}}$

Simplify the numerator.

Rewrite $64$ as $2^{5} \cdot 2$ .

Factor $32$ out of $64$ .

$\frac{\sqrt[5]{32 (2)}}{\sqrt[5]{r}}$

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $4$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the numerator.

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

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

Combine using the product rule for radicals.

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

Move $2$ to the left of $r^{4}$ .

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