Linear Algebra Examples

$\frac{\sqrt[13]{m n} \cdot \sqrt[13]{n^{12}}}{\sqrt[13]{m^{12}}}$

Step 1

Combine $\sqrt[13]{m n}$ and $\sqrt[13]{m^{12}}$ into a single radical.

$\sqrt[13]{\frac{m n}{m^{12}}} \cdot \sqrt[13]{n^{12}}$

Step 2

Reduce the expression $\frac{m n}{m^{12}}$ by cancelling the common factors.

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Step 2.1

Factor $m$ out of $m n$ .

$\sqrt[13]{\frac{m (n)}{m^{12}}} \cdot \sqrt[13]{n^{12}}$

Step 2.2

Factor $m$ out of $m^{12}$ .

$\sqrt[13]{\frac{m (n)}{m \cdot m^{11}}} \cdot \sqrt[13]{n^{12}}$

Step 2.3

Cancel the common factor.

$\sqrt[13]{\frac{m n}{m \cdot m^{11}}} \cdot \sqrt[13]{n^{12}}$

Step 2.4

Rewrite the expression.

$\sqrt[13]{\frac{n}{m^{11}}} \cdot \sqrt[13]{n^{12}}$

Step 3

Rewrite $\sqrt[13]{\frac{n}{m^{11}}}$ as $\frac{\sqrt[13]{n}}{\sqrt[13]{m^{11}}}$ .

$\frac{\sqrt[13]{n}}{\sqrt[13]{m^{11}}} \cdot \sqrt[13]{n^{12}}$

Step 4

Multiply $\frac{\sqrt[13]{n}}{\sqrt[13]{m^{11}}}$ by $\frac{{\sqrt[13]{m^{11}}}^{12}}{{\sqrt[13]{m^{11}}}^{12}}$ .

$\frac{\sqrt[13]{n}}{\sqrt[13]{m^{11}}} \cdot \frac{{\sqrt[13]{m^{11}}}^{12}}{{\sqrt[13]{m^{11}}}^{12}} \cdot \sqrt[13]{n^{12}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt[13]{n}}{\sqrt[13]{m^{11}}}$ by $\frac{{\sqrt[13]{m^{11}}}^{12}}{{\sqrt[13]{m^{11}}}^{12}}$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{\sqrt[13]{m^{11}} {\sqrt[13]{m^{11}}}^{12}} \cdot \sqrt[13]{n^{12}}$

Step 5.2

Raise $\sqrt[13]{m^{11}}$ to the power of $1$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{{\sqrt[13]{m^{11}}}^{1} {\sqrt[13]{m^{11}}}^{12}} \cdot \sqrt[13]{n^{12}}$

Step 5.3

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

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{{\sqrt[13]{m^{11}}}^{1 + 12}} \cdot \sqrt[13]{n^{12}}$

Step 5.4

Add $1$ and $12$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{{\sqrt[13]{m^{11}}}^{13}} \cdot \sqrt[13]{n^{12}}$

Step 5.5

Rewrite ${\sqrt[13]{m^{11}}}^{13}$ as $m^{11}$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[13]{m^{11}}$ as $m^{\frac{11}{13}}$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{{(m^{\frac{11}{13}})}^{13}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.2

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

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{11}{13} \cdot 13}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.3

Combine $\frac{11}{13}$ and $13$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{11 \cdot 13}{13}}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.4

Multiply $11$ by $13$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{143}{13}}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.5

Cancel the common factor of $143$ and $13$ .

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Step 5.5.5.1

Factor $13$ out of $143$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{13 \cdot 11}{13}}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.5.2

Cancel the common factors.

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Step 5.5.5.2.1

Factor $13$ out of $13$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{13 \cdot 11}{13 (1)}}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.5.2.2

Cancel the common factor.

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{13 \cdot 11}{13 \cdot 1}}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.5.2.3

Rewrite the expression.

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{\frac{11}{1}}} \cdot \sqrt[13]{n^{12}}$

Step 5.5.5.2.4

Divide $11$ by $1$ .

$\frac{\sqrt[13]{n} {\sqrt[13]{m^{11}}}^{12}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[13]{m^{11}}}^{12}$ as $\sqrt[13]{{(m^{11})}^{12}}$ .

$\frac{\sqrt[13]{n} \sqrt[13]{{(m^{11})}^{12}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6.2

Multiply the exponents in ${(m^{11})}^{12}$ .

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Step 6.2.1

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

$\frac{\sqrt[13]{n} \sqrt[13]{m^{11 \cdot 12}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6.2.2

Multiply $11$ by $12$ .

$\frac{\sqrt[13]{n} \sqrt[13]{m^{132}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6.3

Rewrite $m^{132}$ as ${(m^{10})}^{13} m^{2}$ .

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Step 6.3.1

Factor out $m^{130}$ .

$\frac{\sqrt[13]{n} \sqrt[13]{m^{130} m^{2}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6.3.2

Rewrite $m^{130}$ as ${(m^{10})}^{13}$ .

$\frac{\sqrt[13]{n} \sqrt[13]{{(m^{10})}^{13} m^{2}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6.4

Pull terms out from under the radical.

$\frac{\sqrt[13]{n} m^{10} \sqrt[13]{m^{2}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 6.5

Combine using the product rule for radicals.

$\frac{m^{10} \sqrt[13]{n m^{2}}}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 7

Cancel the common factor of $m^{10}$ and $m^{11}$ .

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Step 7.1

Factor $m^{10}$ out of $m^{10} \sqrt[13]{n m^{2}}$ .

$\frac{m^{10} (\sqrt[13]{n m^{2}})}{m^{11}} \cdot \sqrt[13]{n^{12}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $m^{10}$ out of $m^{11}$ .

$\frac{m^{10} (\sqrt[13]{n m^{2}})}{m^{10} m} \cdot \sqrt[13]{n^{12}}$

Step 7.2.2

Cancel the common factor.

$\frac{m^{10} \sqrt[13]{n m^{2}}}{m^{10} m} \cdot \sqrt[13]{n^{12}}$

Step 7.2.3

Rewrite the expression.

$\frac{\sqrt[13]{n m^{2}}}{m} \cdot \sqrt[13]{n^{12}}$

Step 8

Multiply $\frac{\sqrt[13]{n m^{2}}}{m} \sqrt[13]{n^{12}}$ .

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Step 8.1

Combine $\frac{\sqrt[13]{n m^{2}}}{m}$ and $\sqrt[13]{n^{12}}$ .

$\frac{\sqrt[13]{n m^{2}} \sqrt[13]{n^{12}}}{m}$

Step 8.2

Combine using the product rule for radicals.

$\frac{\sqrt[13]{n m^{2} n^{12}}}{m}$

Step 8.3

Multiply $n$ by $n^{12}$ by adding the exponents.

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Step 8.3.1

Move $n^{12}$ .

$\frac{\sqrt[13]{n^{12} n m^{2}}}{m}$

Step 8.3.2

Multiply $n^{12}$ by $n$ .

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Step 8.3.2.1

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

$\frac{\sqrt[13]{n^{12} n^{1} m^{2}}}{m}$

Step 8.3.2.2

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

$\frac{\sqrt[13]{n^{12 + 1} m^{2}}}{m}$

Step 8.3.3

Add $12$ and $1$ .

$\frac{\sqrt[13]{n^{13} m^{2}}}{m}$

Step 9

Pull terms out from under the radical.

$\frac{n \sqrt[13]{m^{2}}}{m}$