Simplify fifth root of (5n)/(16m^5)

$\sqrt[5]{\frac{5 n}{16 m^{5}}}$

Step 1

Rewrite $\frac{5 n}{16 m^{5}}$ as ${(\frac{1}{m})}^{5} \frac{5 n}{16}$ .

Factor the perfect power $1^{5}$ out of $5 n$ .

$\sqrt[5]{\frac{1^{5} (5 n)}{16 m^{5}}}$

Factor the perfect power $m^{5}$ out of $16 m^{5}$ .

$\sqrt[5]{\frac{1^{5} (5 n)}{m^{5} \cdot 16}}$

Rearrange the fraction $\frac{1^{5} (5 n)}{m^{5} \cdot 16}$ .

$\sqrt[5]{{(\frac{1}{m})}^{5} \frac{5 n}{16}}$

Step 2

Pull terms out from under the radical.

$\frac{1}{m} \sqrt[5]{\frac{5 n}{16}}$

Step 3

Rewrite $\sqrt[5]{\frac{5 n}{16}}$ as $\frac{\sqrt[5]{5 n}}{\sqrt[5]{16}}$ .

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n}}{\sqrt[5]{16}}$

Step 4

Multiply $\frac{\sqrt[5]{5 n}}{\sqrt[5]{16}}$ by $\frac{{\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{4}}$ .

$\frac{1}{m} (\frac{\sqrt[5]{5 n}}{\sqrt[5]{16}} \cdot \frac{{\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{4}})$

Step 5

Combine fractions.

Combine and simplify the denominator.

Multiply $\frac{\sqrt[5]{5 n}}{\sqrt[5]{16}}$ by $\frac{{\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{4}}$ .

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{\sqrt[5]{16} {\sqrt[5]{16}}^{4}}$

Raise $\sqrt[5]{16}$ to the power of $1$ .

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{1} {\sqrt[5]{16}}^{4}}$

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

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{5}}$

Rewrite ${\sqrt[5]{16}}^{5}$ as $16$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{16}$ as $16^{\frac{1}{5}}$ .

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{{(16^{\frac{1}{5}})}^{5}}$

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

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{16^{\frac{1}{5} \cdot 5}}$

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

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{16^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{16^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{16^{1}}$

Evaluate the exponent.

$\frac{1}{m} \cdot \frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{16}$

Combine.

$\frac{1 (\sqrt[5]{5 n} {\sqrt[5]{16}}^{4})}{m \cdot 16}$

Multiply $\sqrt[5]{5 n}$ by $1$ .

$\frac{\sqrt[5]{5 n} {\sqrt[5]{16}}^{4}}{m \cdot 16}$

Step 6

Simplify the numerator.

Rewrite ${\sqrt[5]{16}}^{4}$ as ${(16^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{5 n} \sqrt[5]{16^{4}}}{m \cdot 16}$

Raise $16$ to the power of $4$ .

$\frac{\sqrt[5]{5 n} \sqrt[5]{65536}}{m \cdot 16}$

Rewrite $65536$ as $8^{5} \cdot 2$ .

Factor $32768$ out of $65536$ .

$\frac{\sqrt[5]{5 n} \sqrt[5]{32768 (2)}}{m \cdot 16}$

Rewrite $32768$ as $8^{5}$ .

$\frac{\sqrt[5]{5 n} \sqrt[5]{8^{5} \cdot 2}}{m \cdot 16}$

Pull terms out from under the radical.

$\frac{\sqrt[5]{5 n} \cdot 8 \sqrt[5]{2}}{m \cdot 16}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{8 \sqrt[5]{5 n \cdot 2}}{m \cdot 16}$

Multiply $2$ by $5$ .

$\frac{8 \sqrt[5]{10 n}}{m \cdot 16}$

Step 7

Reduce the expression by cancelling the common factors.

Cancel the common factor of $8$ and $16$ .

Factor $8$ out of $8 \sqrt[5]{10 n}$ .

$\frac{8 (\sqrt[5]{10 n})}{m \cdot 16}$

Cancel the common factors.

Factor $8$ out of $m \cdot 16$ .

$\frac{8 (\sqrt[5]{10 n})}{8 (m \cdot 2)}$

Cancel the common factor.

$\frac{8 \sqrt[5]{10 n}}{8 (m \cdot 2)}$

Rewrite the expression.

$\frac{\sqrt[5]{10 n}}{m \cdot 2}$

Move $2$ to the left of $m$ .

$\frac{\sqrt[5]{10 n}}{2 m}$

Simplify fifth root of (5n)/(16m^5)

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