Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $3$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $2 x$ .

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

Step 4.3

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

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

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply $3$ by $8$ .

$\frac{\sqrt[4]{24 x^{3}}}{2 x}$