Algebra Examples

$\sqrt[18]{\frac{1}{n^{4} \cdot n^{5}}}$

Step 1

Multiply $n^{4}$ by $n^{5}$ 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.

$\sqrt[18]{\frac{1}{n^{4 + 5}}}$

Step 1.2

Add $4$ and $5$ .

$\sqrt[18]{\frac{1}{n^{9}}}$

Step 2

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

$\sqrt[18]{\frac{1^{3}}{n^{9}}}$

Step 3

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

$\sqrt[18]{\frac{1^{3}}{{(n^{3})}^{3}}}$

Step 4

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

$\sqrt[18]{{(\frac{1}{n^{3}})}^{3}}$

Step 5

Rewrite $\sqrt[18]{{(\frac{1}{n^{3}})}^{3}}$ as $\sqrt[6]{\sqrt[3]{{(\frac{1}{n^{3}})}^{3}}}$ .

$\sqrt[6]{\sqrt[3]{{(\frac{1}{n^{3}})}^{3}}}$

Step 6

Pull terms out from under the radical.

$\sqrt[6]{\frac{1}{n^{3}}}$

Step 7

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

$\sqrt[6]{\frac{1^{3}}{n^{3}}}$

Step 8

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

$\sqrt[6]{{(\frac{1}{n})}^{3}}$

Step 9

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

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

Step 10

Pull terms out from under the radical.

$\sqrt{\frac{1}{n}}$

Step 11

Rewrite $\sqrt{\frac{1}{n}}$ as $\frac{\sqrt{1}}{\sqrt{n}}$ .

$\frac{\sqrt{1}}{\sqrt{n}}$

Step 12

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{n}}$

Step 13

Multiply $\frac{1}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$\frac{1}{\sqrt{n}} \cdot \frac{\sqrt{n}}{\sqrt{n}}$

Step 14

Combine and simplify the denominator.

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Step 14.1

Multiply $\frac{1}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$\frac{\sqrt{n}}{\sqrt{n} \sqrt{n}}$

Step 14.2

Raise $\sqrt{n}$ to the power of $1$ .

$\frac{\sqrt{n}}{{\sqrt{n}}^{1} \sqrt{n}}$

Step 14.3

Raise $\sqrt{n}$ to the power of $1$ .

$\frac{\sqrt{n}}{{\sqrt{n}}^{1} {\sqrt{n}}^{1}}$

Step 14.4

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

$\frac{\sqrt{n}}{{\sqrt{n}}^{1 + 1}}$

Step 14.5

Add $1$ and $1$ .

$\frac{\sqrt{n}}{{\sqrt{n}}^{2}}$

Step 14.6

Rewrite ${\sqrt{n}}^{2}$ as $n$ .

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Step 14.6.1

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

$\frac{\sqrt{n}}{{(n^{\frac{1}{2}})}^{2}}$

Step 14.6.2

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

$\frac{\sqrt{n}}{n^{\frac{1}{2} \cdot 2}}$

Step 14.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{n}}{n^{\frac{2}{2}}}$

Step 14.6.4

Cancel the common factor of $2$ .

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Step 14.6.4.1

Cancel the common factor.

$\frac{\sqrt{n}}{n^{\frac{2}{2}}}$

Step 14.6.4.2

Rewrite the expression.

$\frac{\sqrt{n}}{n^{1}}$

Step 14.6.5

Simplify.

$\frac{\sqrt{n}}{n}$