Algebra Examples

$\sqrt[4]{\frac{81}{4 x^{6}}}$

Step 1

Rewrite $\frac{81}{4 x^{6}}$ as ${(\frac{3}{x})}^{4} \frac{1}{4 x^{2}}$ .

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Step 1.1

Factor the perfect power $3^{4}$ out of $81$ .

$\sqrt[4]{\frac{3^{4} \cdot 1}{4 x^{6}}}$

Step 1.2

Factor the perfect power $x^{4}$ out of $4 x^{6}$ .

$\sqrt[4]{\frac{3^{4} \cdot 1}{x^{4} \cdot (4 x^{2})}}$

Step 1.3

Rearrange the fraction $\frac{3^{4} \cdot 1}{x^{4} \cdot (4 x^{2})}$ .

$\sqrt[4]{{(\frac{3}{x})}^{4} \frac{1}{4 x^{2}}}$

Step 2

Pull terms out from under the radical.

$\frac{3}{x} \sqrt[4]{\frac{1}{4 x^{2}}}$

Step 3

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

$\frac{3}{x} \sqrt[4]{\frac{1^{2}}{4 x^{2}}}$

Step 4

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{3}{x} \sqrt[4]{\frac{1^{2}}{{(2 x)}^{2}}}$

Step 5

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

$\frac{3}{x} \sqrt[4]{{(\frac{1}{2 x})}^{2}}$

Step 6

Rewrite $\sqrt[4]{{(\frac{1}{2 x})}^{2}}$ as $\sqrt{\sqrt{{(\frac{1}{2 x})}^{2}}}$ .

$\frac{3}{x} \sqrt{\sqrt{{(\frac{1}{2 x})}^{2}}}$

Step 7

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

$\frac{3}{x} \sqrt{\frac{1}{2 x}}$

Step 8

Rewrite $\sqrt{\frac{1}{2 x}}$ as $\frac{\sqrt{1}}{\sqrt{2 x}}$ .

$\frac{3}{x} \cdot \frac{\sqrt{1}}{\sqrt{2 x}}$

Step 9

Any root of $1$ is $1$ .

$\frac{3}{x} \cdot \frac{1}{\sqrt{2 x}}$

Step 10

Multiply $\frac{1}{\sqrt{2 x}}$ by $\frac{\sqrt{2 x}}{\sqrt{2 x}}$ .

$\frac{3}{x} (\frac{1}{\sqrt{2 x}} \cdot \frac{\sqrt{2 x}}{\sqrt{2 x}})$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{1}{\sqrt{2 x}}$ by $\frac{\sqrt{2 x}}{\sqrt{2 x}}$ .

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{\sqrt{2 x} \sqrt{2 x}}$

Step 11.2

Raise $\sqrt{2 x}$ to the power of $1$ .

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{\sqrt{2 x}}^{1} \sqrt{2 x}}$

Step 11.3

Raise $\sqrt{2 x}$ to the power of $1$ .

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}}$

Step 11.4

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

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{\sqrt{2 x}}^{1 + 1}}$

Step 11.5

Add $1$ and $1$ .

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{\sqrt{2 x}}^{2}}$

Step 11.6

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

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Step 11.6.1

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

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{({(2 x)}^{\frac{1}{2}})}^{2}}$

Step 11.6.2

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

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{(2 x)}^{\frac{1}{2} \cdot 2}}$

Step 11.6.3

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

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{(2 x)}^{\frac{2}{2}}}$

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{(2 x)}^{\frac{2}{2}}}$

Step 11.6.4.2

Rewrite the expression.

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{{(2 x)}^{1}}$

Step 11.6.5

Simplify.

$\frac{3}{x} \cdot \frac{\sqrt{2 x}}{2 x}$

Step 12

Multiply $\frac{3}{x} \cdot \frac{\sqrt{2 x}}{2 x}$ .

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Step 12.1

Multiply $\frac{3}{x}$ by $\frac{\sqrt{2 x}}{2 x}$ .

$\frac{3 \sqrt{2 x}}{x (2 x)}$

Step 12.2

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

$\frac{3 \sqrt{2 x}}{2 (x^{1} x)}$

Step 12.3

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

$\frac{3 \sqrt{2 x}}{2 (x^{1} x^{1})}$

Step 12.4

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

$\frac{3 \sqrt{2 x}}{2 x^{1 + 1}}$

Step 12.5

Add $1$ and $1$ .

$\frac{3 \sqrt{2 x}}{2 x^{2}}$