Simplify fifth root of (160m^6)/(n^7)

$\sqrt[5]{\frac{160 m^{6}}{n^{7}}}$

Step 1

Rewrite $\frac{160 m^{6}}{n^{7}}$ as ${(\frac{2 m}{n})}^{5} \frac{5 m}{n^{2}}$ .

Factor the perfect power ${(2 m)}^{5}$ out of $160 m^{6}$ .

$\sqrt[5]{\frac{{(2 m)}^{5} \cdot (5 m)}{n^{7}}}$

Factor the perfect power $n^{5}$ out of $n^{7}$ .

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

Rearrange the fraction $\frac{{(2 m)}^{5} \cdot (5 m)}{n^{5} n^{2}}$ .

$\sqrt[5]{{(\frac{2 m}{n})}^{5} \frac{5 m}{n^{2}}}$

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt[5]{\frac{5 m}{n^{2}}}$ as $\frac{\sqrt[5]{5 m}}{\sqrt[5]{n^{2}}}$ .

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

Step 4

Combine.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

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

Move $\sqrt[5]{n^{2}}$ .

$\frac{2 m \sqrt[5]{5 m} {\sqrt[5]{n^{2}}}^{4}}{n (\sqrt[5]{n^{2}} {\sqrt[5]{n^{2}}}^{4})}$

Raise $\sqrt[5]{n^{2}}$ to the power of $1$ .

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

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

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

Add $1$ and $4$ .

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

Rewrite ${\sqrt[5]{n^{2}}}^{5}$ as $n^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{n^{2}}$ as $n^{\frac{2}{5}}$ .

$\frac{2 m \sqrt[5]{5 m} {\sqrt[5]{n^{2}}}^{4}}{n {(n^{\frac{2}{5}})}^{5}}$

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

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

Combine $\frac{2}{5}$ and $5$ .

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

Multiply $2$ by $5$ .

$\frac{2 m \sqrt[5]{5 m} {\sqrt[5]{n^{2}}}^{4}}{n \cdot n^{\frac{10}{5}}}$

Cancel the common factor of $10$ and $5$ .

Factor $5$ out of $10$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{2 m \sqrt[5]{5 m} {\sqrt[5]{n^{2}}}^{4}}{n \cdot n^{\frac{5 \cdot 2}{5 (1)}}}$

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Step 7

Multiply $n$ by $n^{2}$ by adding the exponents.

Multiply $n$ by $n^{2}$ .

Raise $n$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

$\frac{2 m \sqrt[5]{5 m} {\sqrt[5]{n^{2}}}^{4}}{n^{3}}$

Step 8

Simplify the numerator.

Rewrite ${\sqrt[5]{n^{2}}}^{4}$ as ${({(n^{2})}^{4})}^{\frac{1}{5}}$ .

$\frac{2 m \sqrt[5]{5 m} \sqrt[5]{{(n^{2})}^{4}}}{n^{3}}$

Multiply the exponents in ${(n^{2})}^{4}$ .

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

$\frac{2 m \sqrt[5]{5 m} \sqrt[5]{n^{2 \cdot 4}}}{n^{3}}$

Multiply $2$ by $4$ .

$\frac{2 m \sqrt[5]{5 m} \sqrt[5]{n^{8}}}{n^{3}}$

Factor out $n^{5}$ .

$\frac{2 m \sqrt[5]{5 m} \sqrt[5]{n^{5} n^{3}}}{n^{3}}$

Pull terms out from under the radical.

$\frac{2 m \sqrt[5]{5 m} n \sqrt[5]{n^{3}}}{n^{3}}$

Combine using the product rule for radicals.

$\frac{2 m n \sqrt[5]{5 m n^{3}}}{n^{3}}$

Step 9

Cancel the common factor of $n$ and $n^{3}$ .

Factor $n$ out of $2 m n \sqrt[5]{5 m n^{3}}$ .

$\frac{n (2 m \sqrt[5]{5 m n^{3}})}{n^{3}}$

Cancel the common factors.

Factor $n$ out of $n^{3}$ .

$\frac{n (2 m \sqrt[5]{5 m n^{3}})}{n \cdot n^{2}}$

Cancel the common factor.

$\frac{n (2 m \sqrt[5]{5 m n^{3}})}{n \cdot n^{2}}$

Rewrite the expression.

$\frac{2 m \sqrt[5]{5 m n^{3}}}{n^{2}}$

Simplify fifth root of (160m^6)/(n^7)

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