Algebra Examples

$\frac{\sqrt{n^{9}}}{\sqrt[4]{n^{7}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $n^{9}$ as ${(n^{4})}^{2} n$ .

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Step 1.1.1

Factor out $n^{8}$ .

$\frac{\sqrt{n^{8} n}}{\sqrt[4]{n^{7}}}$

Step 1.1.2

Rewrite $n^{8}$ as ${(n^{4})}^{2}$ .

$\frac{\sqrt{{(n^{4})}^{2} n}}{\sqrt[4]{n^{7}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{n^{4} \sqrt{n}}{\sqrt[4]{n^{7}}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor out $n^{4}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor $n$ out of $n^{4} \sqrt{n}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $n$ out of $n \sqrt[4]{n^{3}}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $3$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

Combine $\frac{3}{4}$ and $4$ .

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

Step 5.5.4

Multiply $3$ by $4$ .

$\frac{n^{3} \sqrt{n} {\sqrt[4]{n^{3}}}^{3}}{n^{\frac{12}{4}}}$

Step 5.5.5

Cancel the common factor of $12$ and $4$ .

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Step 5.5.5.1

Factor $4$ out of $12$ .

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

Step 5.5.5.2

Cancel the common factors.

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Step 5.5.5.2.1

Factor $4$ out of $4$ .

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

Step 5.5.5.2.2

Cancel the common factor.

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

Step 5.5.5.2.3

Rewrite the expression.

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

Step 5.5.5.2.4

Divide $3$ by $1$ .

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Divide $\sqrt{n} {\sqrt[4]{n^{3}}}^{3}$ by $1$ .

$\sqrt{n} {\sqrt[4]{n^{3}}}^{3}$

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Multiply $3$ by $3$ .

$\sqrt{n} \sqrt[4]{n^{9}}$

Step 9

Rewrite $n^{9}$ as ${(n^{2})}^{4} n$ .

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Step 9.1

Factor out $n^{8}$ .

$\sqrt{n} \sqrt[4]{n^{8} n}$

Step 9.2

Rewrite $n^{8}$ as ${(n^{2})}^{4}$ .

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

Step 10

Pull terms out from under the radical.

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

Step 11

Multiply $\sqrt{n} (n^{2} \sqrt[4]{n})$ .

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Step 11.1

Rewrite the expression using the least common index of $4$ .

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Step 11.1.1

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

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

Step 11.1.2

Rewrite $n^{\frac{1}{2}}$ as $n^{\frac{2}{4}}$ .

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

Step 11.1.3

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

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

Step 11.2

Combine using the product rule for radicals.

$n^{2} \sqrt[4]{n^{2} n}$

Step 11.3

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

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Step 11.3.1

Multiply $n^{2}$ by $n$ .

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Step 11.3.1.1

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

$n^{2} \sqrt[4]{n^{2} n^{1}}$

Step 11.3.1.2

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

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

Step 11.3.2

Add $2$ and $1$ .

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