Algebra Examples

$\frac{\sqrt[3]{17 y^{2}}}{\sqrt[3]{2 x}}$

Step 1

Multiply to rationalize the numerator.

$\frac{\sqrt[3]{17 y^{2}} {\sqrt[3]{17 y^{2}}}^{2}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Step 2

Simplify.

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Raise $\sqrt[3]{17 y^{2}}$ to the power of $1$ .

$\frac{{\sqrt[3]{17 y^{2}}}^{1} {\sqrt[3]{17 y^{2}}}^{2}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

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

$\frac{{\sqrt[3]{17 y^{2}}}^{1 + 2}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Add $1$ and $2$ .

$\frac{{\sqrt[3]{17 y^{2}}}^{3}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Rewrite ${\sqrt[3]{17 y^{2}}}^{3}$ as $17 y^{2}$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{17 y^{2}}$ as ${(17 y^{2})}^{\frac{1}{3}}$ .

$\frac{{({(17 y^{2})}^{\frac{1}{3}})}^{3}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

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

$\frac{{(17 y^{2})}^{\frac{1}{3} \cdot 3}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

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

$\frac{{(17 y^{2})}^{\frac{3}{3}}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{{(17 y^{2})}^{\frac{3}{3}}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Rewrite the expression.

$\frac{{(17 y^{2})}^{1}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Simplify.

$\frac{17 y^{2}}{\sqrt[3]{2 x} {\sqrt[3]{17 y^{2}}}^{2}}$

Start simplifying.

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{{(17 y^{2})}^{2}}}$

Apply the product rule to $17 y^{2}$ .

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{17^{2} {(y^{2})}^{2}}}$

Raise $17$ to the power of $2$ .

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{289 {(y^{2})}^{2}}}$

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

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

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{289 y^{2 \cdot 2}}}$

Multiply $2$ by $2$ .

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{289 y^{4}}}$

Rewrite $289 y^{4}$ as $y^{3} \cdot (289 y)$ .

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Factor out $y^{3}$ .

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{289 (y^{3} y)}}$

Reorder $289$ and $y^{3}$ .

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{y^{3} \cdot 289 y}}$

Add parentheses.

$\frac{17 y^{2}}{\sqrt[3]{2 x} \sqrt[3]{y^{3} \cdot (289 y)}}$

Pull terms out from under the radical.

$\frac{17 y^{2}}{\sqrt[3]{2 x} (y \sqrt[3]{289 y})}$

Multiply $\sqrt[3]{2 x} (y \sqrt[3]{289 y})$ .

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Combine using the product rule for radicals.

$\frac{17 y^{2}}{y \sqrt[3]{2 x (289 y)}}$

Multiply $289$ by $2$ .

$\frac{17 y^{2}}{y \sqrt[3]{578 x y}}$