Algebra Examples

$\sqrt[24]{\frac{1}{n^{7} \cdot n^{2}}}$

Step 1

Multiply $n^{7}$ by $n^{2}$ by adding the exponents.

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Step 1.1

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

$\sqrt[24]{\frac{1}{n^{7 + 2}}}$

Step 1.2

Add $7$ and $2$ .

$\sqrt[24]{\frac{1}{n^{9}}}$

Step 2

Rewrite $1$ as $1^{3}$ .

$\sqrt[24]{\frac{1^{3}}{n^{9}}}$

Step 3

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

$\sqrt[24]{\frac{1^{3}}{{(n^{3})}^{3}}}$

Step 4

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

$\sqrt[24]{{(\frac{1}{n^{3}})}^{3}}$

Step 5

Rewrite $\sqrt[24]{{(\frac{1}{n^{3}})}^{3}}$ as $\sqrt[8]{\sqrt[3]{{(\frac{1}{n^{3}})}^{3}}}$ .

$\sqrt[8]{\sqrt[3]{{(\frac{1}{n^{3}})}^{3}}}$

Step 6

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

$\sqrt[8]{\frac{1}{n^{3}}}$

Step 7

Rewrite $\sqrt[8]{\frac{1}{n^{3}}}$ as $\frac{\sqrt[8]{1}}{\sqrt[8]{n^{3}}}$ .

$\frac{\sqrt[8]{1}}{\sqrt[8]{n^{3}}}$

Step 8

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[8]{n^{3}}}$

Step 9

Multiply $\frac{1}{\sqrt[8]{n^{3}}}$ by $\frac{{\sqrt[8]{n^{3}}}^{7}}{{\sqrt[8]{n^{3}}}^{7}}$ .

$\frac{1}{\sqrt[8]{n^{3}}} \cdot \frac{{\sqrt[8]{n^{3}}}^{7}}{{\sqrt[8]{n^{3}}}^{7}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{\sqrt[8]{n^{3}}}$ by $\frac{{\sqrt[8]{n^{3}}}^{7}}{{\sqrt[8]{n^{3}}}^{7}}$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{\sqrt[8]{n^{3}} {\sqrt[8]{n^{3}}}^{7}}$

Step 10.2

Raise $\sqrt[8]{n^{3}}$ to the power of $1$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{{\sqrt[8]{n^{3}}}^{1} {\sqrt[8]{n^{3}}}^{7}}$

Step 10.3

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

$\frac{{\sqrt[8]{n^{3}}}^{7}}{{\sqrt[8]{n^{3}}}^{1 + 7}}$

Step 10.4

Add $1$ and $7$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{{\sqrt[8]{n^{3}}}^{8}}$

Step 10.5

Rewrite ${\sqrt[8]{n^{3}}}^{8}$ as $n^{3}$ .

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Step 10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{n^{3}}$ as $n^{\frac{3}{8}}$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{{(n^{\frac{3}{8}})}^{8}}$

Step 10.5.2

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

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{3}{8} \cdot 8}}$

Step 10.5.3

Combine $\frac{3}{8}$ and $8$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{3 \cdot 8}{8}}}$

Step 10.5.4

Multiply $3$ by $8$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{24}{8}}}$

Step 10.5.5

Cancel the common factor of $24$ and $8$ .

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Step 10.5.5.1

Factor $8$ out of $24$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{8 \cdot 3}{8}}}$

Step 10.5.5.2

Cancel the common factors.

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Step 10.5.5.2.1

Factor $8$ out of $8$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{8 \cdot 3}{8 (1)}}}$

Step 10.5.5.2.2

Cancel the common factor.

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{8 \cdot 3}{8 \cdot 1}}}$

Step 10.5.5.2.3

Rewrite the expression.

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{\frac{3}{1}}}$

Step 10.5.5.2.4

Divide $3$ by $1$ .

$\frac{{\sqrt[8]{n^{3}}}^{7}}{n^{3}}$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite ${\sqrt[8]{n^{3}}}^{7}$ as $\sqrt[8]{{(n^{3})}^{7}}$ .

$\frac{\sqrt[8]{{(n^{3})}^{7}}}{n^{3}}$

Step 11.2

Multiply the exponents in ${(n^{3})}^{7}$ .

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Step 11.2.1

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

$\frac{\sqrt[8]{n^{3 \cdot 7}}}{n^{3}}$

Step 11.2.2

Multiply $3$ by $7$ .

$\frac{\sqrt[8]{n^{21}}}{n^{3}}$

Step 11.3

Rewrite $n^{21}$ as ${(n^{2})}^{8} n^{5}$ .

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Step 11.3.1

Factor out $n^{16}$ .

$\frac{\sqrt[8]{n^{16} n^{5}}}{n^{3}}$

Step 11.3.2

Rewrite $n^{16}$ as ${(n^{2})}^{8}$ .

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

Step 11.4

Pull terms out from under the radical.

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

Step 12

Cancel the common factor of $n^{2}$ and $n^{3}$ .

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Step 12.1

Factor $n^{2}$ out of $n^{2} \sqrt[8]{n^{5}}$ .

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

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $n^{2}$ out of $n^{3}$ .

$\frac{n^{2} (\sqrt[8]{n^{5}})}{n^{2} n}$

Step 12.2.2

Cancel the common factor.

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

Step 12.2.3

Rewrite the expression.

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