Algebra Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $1$ and $2$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $3$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Simplify.

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

Step 2.5

Start simplifying.

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

Multiply $2$ by $2$ .

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

Step 2.9

Rewrite $4 y^{4}$ as $y^{3} \cdot (4 y)$ .

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Step 2.9.1

Factor out $y^{3}$ .

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

Step 2.9.2

Reorder $4$ and $y^{3}$ .

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

Step 2.9.3

Add parentheses.

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

Step 2.10

Pull terms out from under the radical.

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

Step 2.11

Multiply $\sqrt[3]{3 x} (y \sqrt[3]{4 y})$ .

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Step 2.11.1

Combine using the product rule for radicals.

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

Step 2.11.2

Multiply $4$ by $3$ .

$\frac{2 y^{2}}{y \sqrt[3]{12 x y}}$