Precalculus Examples

$\frac{\sqrt{\sqrt[3]{4 m^{4} n^{5}}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4 m^{4} n^{5}$ as ${(m n)}^{3} \cdot (4 m n^{2})$ .

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Step 1.1.1

Factor out $m^{3}$ .

$\frac{\sqrt{\sqrt[3]{4 (m^{3} m) n^{5}}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.1.2

Factor out $n^{3}$ .

$\frac{\sqrt{\sqrt[3]{4 (m^{3} m) (n^{3} n^{2})}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.1.3

Move $m$ .

$\frac{\sqrt{\sqrt[3]{4 (m^{3}) n^{3} m n^{2}}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.1.4

Move $4$ .

$\frac{\sqrt{\sqrt[3]{(m^{3}) n^{3} \cdot 4 m n^{2}}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.1.5

Rewrite $(m^{3}) n^{3}$ as ${(m n)}^{3}$ .

$\frac{\sqrt{\sqrt[3]{{(m n)}^{3} \cdot 4 m n^{2}}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.1.6

Add parentheses.

$\frac{\sqrt{\sqrt[3]{{(m n)}^{3} \cdot 4 (m n^{2})}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.1.7

Add parentheses.

$\frac{\sqrt{\sqrt[3]{{(m n)}^{3} \cdot (4 m n^{2})}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}}}{\sqrt[3]{256 m^{10} n^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $256 m^{10} n^{2}$ as ${(4 m^{3})}^{3} \cdot (4 m n^{2})$ .

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Step 2.1.1

Factor $64$ out of $256$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}}}{\sqrt[3]{64 (4) m^{10} n^{2}}}$

Step 2.1.2

Rewrite $64$ as $4^{3}$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}}}{\sqrt[3]{4^{3} \cdot 4 m^{10} n^{2}}}$

Step 2.1.3

Factor out $m^{9}$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}}}{\sqrt[3]{4^{3} \cdot 4 (m^{9} m) n^{2}}}$

Step 2.1.4

Rewrite $m^{9}$ as ${(m^{3})}^{3}$ .

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

Step 2.1.5

Move $4$ .

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

Step 2.1.6

Rewrite $4^{3} {(m^{3})}^{3}$ as ${(4 m^{3})}^{3}$ .

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

Step 2.1.7

Add parentheses.

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

Step 2.1.8

Add parentheses.

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Multiply $\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}}}{4 m^{3} \sqrt[3]{4 m n^{2}}}$ by $\frac{{\sqrt[3]{4 m n^{2}}}^{2}}{{\sqrt[3]{4 m n^{2}}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}}}{4 m^{3} \sqrt[3]{4 m n^{2}}}$ by $\frac{{\sqrt[3]{4 m n^{2}}}^{2}}{{\sqrt[3]{4 m n^{2}}}^{2}}$ .

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

Step 4.2

Move $\sqrt[3]{4 m n^{2}}$ .

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

Step 4.3

Raise $\sqrt[3]{4 m n^{2}}$ to the power of $1$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 m^{3} ({\sqrt[3]{4 m n^{2}}}^{1} {\sqrt[3]{4 m n^{2}}}^{2})}$

Step 4.4

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

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 m^{3} {\sqrt[3]{4 m n^{2}}}^{1 + 2}}$

Step 4.5

Add $1$ and $2$ .

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

Step 4.6

Rewrite ${\sqrt[3]{4 m n^{2}}}^{3}$ as $4 m n^{2}$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 m n^{2}}$ as ${(4 m n^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 m^{3} {({(4 m n^{2})}^{\frac{1}{3}})}^{3}}$

Step 4.6.2

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

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 m^{3} {(4 m n^{2})}^{\frac{1}{3} \cdot 3}}$

Step 4.6.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 4.6.4

Cancel the common factor of $3$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 m^{3} {(4 m n^{2})}^{1}}$

Step 4.6.5

Simplify.

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

Step 5

Multiply $m^{3}$ by $m$ by adding the exponents.

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Step 5.1

Move $m$ .

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

Step 5.2

Multiply $m$ by $m^{3}$ .

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Step 5.2.1

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

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 (m^{1} m^{3}) (4 n^{2})}$

Step 5.2.2

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

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} {\sqrt[3]{4 m n^{2}}}^{2}}{4 m^{1 + 3} (4 n^{2})}$

Step 5.3

Add $1$ and $3$ .

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[3]{4 m n^{2}}}^{2}$ as $\sqrt[3]{{(4 m n^{2})}^{2}}$ .

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

Step 6.2

Apply the product rule to $4 m n^{2}$ .

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

Step 6.3

Apply the product rule to $4 m$ .

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

Step 6.4

Raise $4$ to the power of $2$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} \sqrt[3]{16 m^{2} {(n^{2})}^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.5

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

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Step 6.5.1

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

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} \sqrt[3]{16 m^{2} n^{2 \cdot 2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.5.2

Multiply $2$ by $2$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} \sqrt[3]{16 m^{2} n^{4}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.6

Rewrite $16 m^{2} n^{4}$ as ${(2 n)}^{3} \cdot (2 m^{2} n)$ .

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Step 6.6.1

Factor $8$ out of $16$ .

$\frac{\sqrt{m n \sqrt[3]{4 m n^{2}}} \sqrt[3]{8 (2) m^{2} n^{4}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.6.2

Rewrite $8$ as $2^{3}$ .

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

Step 6.6.3

Factor out $n^{3}$ .

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

Step 6.6.4

Move $m^{2}$ .

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

Step 6.6.5

Move $2$ .

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

Step 6.6.6

Rewrite $2^{3} n^{3}$ as ${(2 n)}^{3}$ .

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

Step 6.6.7

Add parentheses.

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

Step 6.6.8

Add parentheses.

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

Step 6.7

Pull terms out from under the radical.

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

Step 6.8

Combine exponents.

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Step 6.8.1

Rewrite the expression using the least common index of $6$ .

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Step 6.8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m n \sqrt[3]{4 m n^{2}}}$ as ${(m n \sqrt[3]{4 m n^{2}})}^{\frac{1}{2}}$ .

$\frac{2 n ({(m n \sqrt[3]{4 m n^{2}})}^{\frac{1}{2}} \sqrt[3]{2 m^{2} n})}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.1.2

Rewrite ${(m n \sqrt[3]{4 m n^{2}})}^{\frac{1}{2}}$ as ${(m n \sqrt[3]{4 m n^{2}})}^{\frac{3}{6}}$ .

$\frac{2 n ({(m n \sqrt[3]{4 m n^{2}})}^{\frac{3}{6}} \sqrt[3]{2 m^{2} n})}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.1.3

Rewrite ${(m n \sqrt[3]{4 m n^{2}})}^{\frac{3}{6}}$ as $\sqrt[6]{{(m n \sqrt[3]{4 m n^{2}})}^{3}}$ .

$\frac{2 n (\sqrt[6]{{(m n \sqrt[3]{4 m n^{2}})}^{3}} \sqrt[3]{2 m^{2} n})}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 m^{2} n}$ as ${(2 m^{2} n)}^{\frac{1}{3}}$ .

$\frac{2 n (\sqrt[6]{{(m n \sqrt[3]{4 m n^{2}})}^{3}} {(2 m^{2} n)}^{\frac{1}{3}})}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.1.5

Rewrite ${(2 m^{2} n)}^{\frac{1}{3}}$ as ${(2 m^{2} n)}^{\frac{2}{6}}$ .

$\frac{2 n (\sqrt[6]{{(m n \sqrt[3]{4 m n^{2}})}^{3}} {(2 m^{2} n)}^{\frac{2}{6}})}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.1.6

Rewrite ${(2 m^{2} n)}^{\frac{2}{6}}$ as $\sqrt[6]{{(2 m^{2} n)}^{2}}$ .

$\frac{2 n (\sqrt[6]{{(m n \sqrt[3]{4 m n^{2}})}^{3}} \sqrt[6]{{(2 m^{2} n)}^{2}})}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.2

Combine using the product rule for radicals.

$\frac{2 n \sqrt[6]{{(m n \sqrt[3]{4 m n^{2}})}^{3} {(2 m^{2} n)}^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.3

Apply the product rule to $m n \sqrt[3]{4 m n^{2}}$ .

$\frac{2 n \sqrt[6]{{(m n)}^{3} {\sqrt[3]{4 m n^{2}}}^{3} {(2 m^{2} n)}^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.4

Apply the product rule to $2 m^{2} n$ .

$\frac{2 n \sqrt[6]{{(m n)}^{3} {\sqrt[3]{4 m n^{2}}}^{3} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.5

Apply the product rule to $m n$ .

$\frac{2 n \sqrt[6]{m^{3} n^{3} {\sqrt[3]{4 m n^{2}}}^{3} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.6

Rewrite ${\sqrt[3]{4 m n^{2}}}^{3}$ as $4 m n^{2}$ .

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Step 6.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 m n^{2}}$ as ${(4 m n^{2})}^{\frac{1}{3}}$ .

$\frac{2 n \sqrt[6]{m^{3} n^{3} {({(4 m n^{2})}^{\frac{1}{3}})}^{3} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.6.2

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

$\frac{2 n \sqrt[6]{m^{3} n^{3} {(4 m n^{2})}^{\frac{1}{3} \cdot 3} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.6.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{2 n \sqrt[6]{m^{3} n^{3} {(4 m n^{2})}^{\frac{3}{3}} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.6.4

Cancel the common factor of $3$ .

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Step 6.8.6.4.1

Cancel the common factor.

$\frac{2 n \sqrt[6]{m^{3} n^{3} {(4 m n^{2})}^{\frac{3}{3}} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.6.4.2

Rewrite the expression.

$\frac{2 n \sqrt[6]{m^{3} n^{3} {(4 m n^{2})}^{1} {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.6.5

Simplify.

$\frac{2 n \sqrt[6]{m^{3} n^{3} (4 m n^{2}) {(2 m^{2})}^{2} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.7

Apply the product rule to $2 m^{2}$ .

$\frac{2 n \sqrt[6]{m^{3} n^{3} (4 m n^{2}) (2^{2} {(m^{2})}^{2}) n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.8

Raise $2$ to the power of $2$ .

$\frac{2 n \sqrt[6]{m^{3} n^{3} (4 m n^{2}) (4 {(m^{2})}^{2}) n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.9

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

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Step 6.8.9.1

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

$\frac{2 n \sqrt[6]{m^{3} n^{3} (4 m n^{2}) (4 m^{2 \cdot 2}) n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.9.2

Multiply $2$ by $2$ .

$\frac{2 n \sqrt[6]{m^{3} n^{3} (4 m n^{2}) (4 m^{4}) n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

$\frac{2 n \sqrt[6]{m^{3} n^{3} \cdot 4 m n^{2} \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10

Combine exponents.

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Step 6.8.10.1

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

$\frac{2 n \sqrt[6]{m^{1} m^{3} n^{3} \cdot 4 n^{2} \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.2

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

$\frac{2 n \sqrt[6]{m^{1 + 3} n^{3} \cdot 4 n^{2} \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.3

Add $1$ and $3$ .

$\frac{2 n \sqrt[6]{m^{4} n^{3} \cdot 4 n^{2} \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.4

Multiply $n^{3}$ by $n^{2}$ by adding the exponents.

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Step 6.8.10.4.1

Move $n^{2}$ .

$\frac{2 n \sqrt[6]{m^{4} (n^{2} n^{3}) \cdot 4 \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.4.2

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

$\frac{2 n \sqrt[6]{m^{4} n^{2 + 3} \cdot 4 \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.4.3

Add $2$ and $3$ .

$\frac{2 n \sqrt[6]{m^{4} n^{5} \cdot 4 \cdot 4 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.5

Multiply $4$ by $4$ .

$\frac{2 n \sqrt[6]{m^{4} n^{5} \cdot 16 m^{4} n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.6

Multiply $m^{4}$ by $m^{4}$ by adding the exponents.

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Step 6.8.10.6.1

Move $m^{4}$ .

$\frac{2 n \sqrt[6]{m^{4} m^{4} n^{5} \cdot 16 n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.6.2

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

$\frac{2 n \sqrt[6]{m^{4 + 4} n^{5} \cdot 16 n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.6.3

Add $4$ and $4$ .

$\frac{2 n \sqrt[6]{m^{8} n^{5} \cdot 16 n^{2}}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.7

Multiply $n^{5}$ by $n^{2}$ by adding the exponents.

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Step 6.8.10.7.1

Move $n^{2}$ .

$\frac{2 n \sqrt[6]{m^{8} (n^{2} n^{5}) \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.7.2

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

$\frac{2 n \sqrt[6]{m^{8} n^{2 + 5} \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.8.10.7.3

Add $2$ and $5$ .

$\frac{2 n \sqrt[6]{m^{8} n^{7} \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.9

Rewrite $m^{8} n^{7} \cdot 16$ as ${(m n)}^{6} (m^{2} n \cdot 16)$ .

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Step 6.9.1

Factor out $m^{6}$ .

$\frac{2 n \sqrt[6]{m^{6} m^{2} n^{7} \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.9.2

Factor out $n^{6}$ .

$\frac{2 n \sqrt[6]{m^{6} m^{2} (n^{6} n) \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.9.3

Move $m^{2}$ .

$\frac{2 n \sqrt[6]{m^{6} n^{6} m^{2} n \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.9.4

Rewrite $m^{6} n^{6}$ as ${(m n)}^{6}$ .

$\frac{2 n \sqrt[6]{{(m n)}^{6} m^{2} n \cdot 16}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.9.5

Add parentheses.

$\frac{2 n \sqrt[6]{{(m n)}^{6} m^{2} (n \cdot 16)}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.9.6

Add parentheses.

$\frac{2 n \sqrt[6]{{(m n)}^{6} (m^{2} n \cdot 16)}}{4 m^{4} \cdot 4 n^{2}}$

Step 6.10

Pull terms out from under the radical.

$\frac{2 n (m n \sqrt[6]{m^{2} n \cdot 16})}{4 m^{4} \cdot 4 n^{2}}$

Step 7

Multiply $4$ by $4$ by adding the exponents.

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Step 7.1

Move $4$ .

$\frac{2 n (m n \sqrt[6]{m^{2} n \cdot 16})}{4 \cdot 4 m^{4} n^{2}}$

Step 7.2

Multiply $4$ by $4$ .

$\frac{2 n (m n \sqrt[6]{m^{2} n \cdot 16})}{4^{2} m^{4} n^{2}}$

Step 8

Simplify the numerator.

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Step 8.1

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

$\frac{2 (m (n^{1} n) \sqrt[6]{m^{2} n \cdot 16})}{4^{2} m^{4} n^{2}}$

Step 8.2

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

$\frac{2 (m (n^{1} n^{1}) \sqrt[6]{m^{2} n \cdot 16})}{4^{2} m^{4} n^{2}}$

Step 8.3

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

$\frac{2 (m n^{1 + 1} \sqrt[6]{m^{2} n \cdot 16})}{4^{2} m^{4} n^{2}}$

Step 8.4

Add $1$ and $1$ .

$\frac{2 (m n^{2} \sqrt[6]{m^{2} n \cdot 16})}{4^{2} m^{4} n^{2}}$

Step 9

Raise $4$ to the power of $2$ .

$\frac{2 (m n^{2} \sqrt[6]{m^{2} n \cdot 16})}{16 m^{4} n^{2}}$

Step 10

Remove parentheses.

$\frac{2 m n^{2} \sqrt[6]{m^{2} n \cdot 16}}{16 m^{4} n^{2}}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Cancel the common factor of $2$ and $16$ .

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Step 11.1.1

Factor $2$ out of $2 m n^{2} \sqrt[6]{m^{2} n \cdot 16}$ .

$\frac{2 (m n^{2} \sqrt[6]{m^{2} n \cdot 16})}{16 m^{4} n^{2}}$

Step 11.1.2

Cancel the common factors.

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Step 11.1.2.1

Factor $2$ out of $16 m^{4} n^{2}$ .

$\frac{2 (m n^{2} \sqrt[6]{m^{2} n \cdot 16})}{2 (8 m^{4} n^{2})}$

Step 11.1.2.2

Cancel the common factor.

$\frac{2 (m n^{2} \sqrt[6]{m^{2} n \cdot 16})}{2 (8 m^{4} n^{2})}$

Step 11.1.2.3

Rewrite the expression.

$\frac{m n^{2} \sqrt[6]{m^{2} n \cdot 16}}{8 m^{4} n^{2}}$

Step 11.2

Cancel the common factor of $m$ and $m^{4}$ .

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Step 11.2.1

Factor $m$ out of $m n^{2} \sqrt[6]{m^{2} n \cdot 16}$ .

$\frac{m (n^{2} \sqrt[6]{m^{2} n \cdot 16})}{8 m^{4} n^{2}}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $m$ out of $8 m^{4} n^{2}$ .

$\frac{m (n^{2} \sqrt[6]{m^{2} n \cdot 16})}{m (8 m^{3} n^{2})}$

Step 11.2.2.2

Cancel the common factor.

$\frac{m (n^{2} \sqrt[6]{m^{2} n \cdot 16})}{m (8 m^{3} n^{2})}$

Step 11.2.2.3

Rewrite the expression.

$\frac{n^{2} \sqrt[6]{m^{2} n \cdot 16}}{8 m^{3} n^{2}}$

Step 11.3

Cancel the common factor of $n^{2}$ .

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Step 11.3.1

Cancel the common factor.

$\frac{n^{2} \sqrt[6]{m^{2} n \cdot 16}}{8 m^{3} n^{2}}$

Step 11.3.2

Rewrite the expression.

$\frac{\sqrt[6]{m^{2} n \cdot 16}}{8 m^{3}}$

Step 11.4

Reorder factors in $\frac{\sqrt[6]{m^{2} n \cdot 16}}{8 m^{3}}$ .

$\frac{\sqrt[6]{16 m^{2} n}}{8 m^{3}}$