Algebra Examples

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

Step 1

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

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

Step 2

Multiply to rationalize the numerator.

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

Step 3

Simplify.

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Step 3.1

Raise $\sqrt[3]{2 a^{4}}$ to the power of $1$ .

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

Step 3.2

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

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

Step 3.3

Add $1$ and $2$ .

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

Step 3.4

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

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 a^{4}}$ as ${(2 a^{4})}^{\frac{1}{3}}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 3.4.4

Cancel the common factor of $3$ .

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Step 3.4.4.1

Cancel the common factor.

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

Step 3.4.4.2

Rewrite the expression.

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

Step 3.4.5

Simplify.

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

Step 3.5

Start simplifying.

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

Step 3.6

Apply the product rule to $2 a^{4}$ .

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

Step 3.7

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

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

Step 3.8

Multiply the exponents in ${(a^{4})}^{2}$ .

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Step 3.8.1

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

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

Step 3.8.2

Multiply $4$ by $2$ .

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

Step 3.9

Rewrite $4 a^{8}$ as ${(a^{2})}^{3} \cdot (4 a^{2})$ .

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Step 3.9.1

Factor out $a^{6}$ .

$\frac{2 a^{4}}{\sqrt[3]{3 b} \sqrt[3]{4 (a^{6} a^{2})}}$

Step 3.9.2

Rewrite $a^{6}$ as ${(a^{2})}^{3}$ .

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

Step 3.9.3

Reorder $4$ and ${(a^{2})}^{3}$ .

$\frac{2 a^{4}}{\sqrt[3]{3 b} \sqrt[3]{{(a^{2})}^{3} \cdot 4 a^{2}}}$

Step 3.9.4

Add parentheses.

$\frac{2 a^{4}}{\sqrt[3]{3 b} \sqrt[3]{{(a^{2})}^{3} \cdot (4 a^{2})}}$

Step 3.10

Pull terms out from under the radical.

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

Step 3.11

Multiply $\sqrt[3]{3 b} (a^{2} \sqrt[3]{4 a^{2}})$ .

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Step 3.11.1

Combine using the product rule for radicals.

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

Step 3.11.2

Multiply $4$ by $3$ .

$\frac{2 a^{4}}{a^{2} \sqrt[3]{12 b a^{2}}}$