Algebra Examples

$\sqrt[36]{\frac{m^{- 8}}{m^{4}}}$

Step 1

Reduce the expression $\frac{m^{- 8}}{m^{4}}$ by cancelling the common factors.

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Step 1.1

Multiply by $1$ .

$\sqrt[36]{\frac{m^{- 8} \cdot 1}{m^{4}}}$

Step 1.2

Factor $m^{- 8}$ out of $m^{4}$ .

$\sqrt[36]{\frac{m^{- 8} \cdot 1}{m^{- 8} m^{12}}}$

Step 1.3

Cancel the common factor.

$\sqrt[36]{\frac{m^{- 8} \cdot 1}{m^{- 8} m^{12}}}$

Step 1.4

Rewrite the expression.

$\sqrt[36]{\frac{1}{m^{12}}}$

Step 2

Simplify the expression.

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Step 2.1

Rewrite $1$ as $1^{6}$ .

$\sqrt[36]{\frac{1^{6}}{m^{12}}}$

Step 2.2

Rewrite $m^{12}$ as ${(m^{2})}^{6}$ .

$\sqrt[36]{\frac{1^{6}}{{(m^{2})}^{6}}}$

Step 3

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

$\sqrt[36]{{(\frac{1}{m^{2}})}^{6}}$

Step 4

Rewrite $\sqrt[36]{{(\frac{1}{m^{2}})}^{6}}$ as $\sqrt[6]{\sqrt[6]{{(\frac{1}{m^{2}})}^{6}}}$ .

$\sqrt[6]{\sqrt[6]{{(\frac{1}{m^{2}})}^{6}}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[6]{\frac{1}{m^{2}}}$

Step 6

Rewrite $1$ as $1^{2}$ .

$\sqrt[6]{\frac{1^{2}}{m^{2}}}$

Step 7

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

$\sqrt[6]{{(\frac{1}{m})}^{2}}$

Step 8

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

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

Step 9

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[3]{\frac{1}{m}}$

Step 10

Rewrite $\sqrt[3]{\frac{1}{m}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{m}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{m}}$

Step 11

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{m}}$

Step 12

Multiply $\frac{1}{\sqrt[3]{m}}$ by $\frac{{\sqrt[3]{m}}^{2}}{{\sqrt[3]{m}}^{2}}$ .

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

Step 13

Combine and simplify the denominator.

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Step 13.1

Multiply $\frac{1}{\sqrt[3]{m}}$ by $\frac{{\sqrt[3]{m}}^{2}}{{\sqrt[3]{m}}^{2}}$ .

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

Add $1$ and $2$ .

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

Step 13.5

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

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Step 13.5.1

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

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

Step 13.5.2

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

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

Step 13.5.3

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

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

Step 13.5.4

Cancel the common factor of $3$ .

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Step 13.5.4.1

Cancel the common factor.

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

Step 13.5.4.2

Rewrite the expression.

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

Step 13.5.5

Simplify.

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

Step 14

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

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