Algebra Examples

$\sqrt[4]{\frac{9}{2 x^{2}}}$

Step 1

Rewrite $\sqrt[4]{\frac{9}{2 x^{2}}}$ as $\frac{\sqrt[4]{9}}{\sqrt[4]{2 x^{2}}}$ .

$\frac{\sqrt[4]{9}}{\sqrt[4]{2 x^{2}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.2

Rewrite $\sqrt[4]{3^{2}}$ as $\sqrt{\sqrt{3^{2}}}$ .

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

Step 2.3

Pull terms out from under the radical.

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

Step 2.4

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 3

Multiply $\frac{\sqrt{3}}{\sqrt[4]{2 x^{2}}}$ by $\frac{{\sqrt[4]{2 x^{2}}}^{3}}{{\sqrt[4]{2 x^{2}}}^{3}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{3}}{\sqrt[4]{2 x^{2}}}$ by $\frac{{\sqrt[4]{2 x^{2}}}^{3}}{{\sqrt[4]{2 x^{2}}}^{3}}$ .

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

Step 4.2

Raise $\sqrt[4]{2 x^{2}}$ to the power of $1$ .

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

Step 4.3

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

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

Step 4.4

Add $1$ and $3$ .

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

Step 4.5

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

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2 x^{2}}$ as ${(2 x^{2})}^{\frac{1}{4}}$ .

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

Step 4.5.2

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

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

Step 4.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 4.5.4

Cancel the common factor of $4$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[4]{2 x^{2}}}^{3}$ as $\sqrt[4]{{(2 x^{2})}^{3}}$ .

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

Step 5.2

Apply the product rule to $2 x^{2}$ .

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

Step 5.3

Raise $2$ to the power of $3$ .

$\frac{\sqrt{3} \sqrt[4]{8 {(x^{2})}^{3}}}{2 x^{2}}$

Step 5.4

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 5.4.1

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

$\frac{\sqrt{3} \sqrt[4]{8 x^{2 \cdot 3}}}{2 x^{2}}$

Step 5.4.2

Multiply $2$ by $3$ .

$\frac{\sqrt{3} \sqrt[4]{8 x^{6}}}{2 x^{2}}$

Step 5.5

Rewrite $8 x^{6}$ as $x^{4} \cdot (8 x^{2})$ .

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Step 5.5.1

Factor out $x^{4}$ .

$\frac{\sqrt{3} \sqrt[4]{8 (x^{4} x^{2})}}{2 x^{2}}$

Step 5.5.2

Reorder $8$ and $x^{4}$ .

$\frac{\sqrt{3} \sqrt[4]{x^{4} \cdot 8 x^{2}}}{2 x^{2}}$

Step 5.5.3

Add parentheses.

$\frac{\sqrt{3} \sqrt[4]{x^{4} \cdot (8 x^{2})}}{2 x^{2}}$

Step 5.6

Pull terms out from under the radical.

$\frac{\sqrt{3} | x | \sqrt[4]{8 x^{2}}}{2 x^{2}}$

Step 5.7

Combine exponents.

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Step 5.7.1

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

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Step 5.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{8 x^{2}} \cdot 3^{\frac{1}{2}} | x |}{2 x^{2}}$

Step 5.7.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{8 x^{2}} \cdot 3^{\frac{2}{4}} | x |}{2 x^{2}}$

Step 5.7.1.3

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$\frac{\sqrt[4]{8 x^{2}} \sqrt[4]{3^{2}} | x |}{2 x^{2}}$

Step 5.7.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{8 x^{2} \cdot 3^{2}} | x |}{2 x^{2}}$

Step 5.7.3

Raise $3$ to the power of $2$ .

$\frac{\sqrt[4]{8 x^{2} \cdot 9} | x |}{2 x^{2}}$

Step 5.7.4

Multiply $9$ by $8$ .

$\frac{\sqrt[4]{72 x^{2}} | x |}{2 x^{2}}$