Algebra Examples

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

Step 1

Combine $\sqrt[4]{2 x}$ and $\sqrt[4]{3 x^{2}}$ into a single radical.

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

Step 2

Reduce the expression $\frac{2 x}{3 x^{2}}$ by cancelling the common factors.

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Step 2.1

Factor $x$ out of $2 x$ .

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

Step 2.2

Factor $x$ out of $3 x^{2}$ .

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

Step 2.3

Cancel the common factor.

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

Step 2.4

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $3$ .

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

Step 5.5

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

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 x}$ as ${(3 x)}^{\frac{1}{4}}$ .

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $4$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

Apply the product rule to $3 x$ .

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

Step 6.3

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

$\frac{\sqrt[4]{2} \sqrt[4]{27 x^{3}}}{3 x}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt[4]{2 (27 x^{3})}}{3 x}$

Step 7.2

Multiply $2$ by $27$ .

$\frac{\sqrt[4]{54 x^{3}}}{3 x}$