Algebra Examples

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

Step 1

Rewrite $\frac{x}{y^{8}}$ as ${(\frac{1}{y^{2}})}^{3} \frac{x}{y^{2}}$ .

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Step 1.1

Factor the perfect power $1^{3}$ out of $x$ .

$\sqrt[3]{\frac{1^{3} x}{y^{8}}}$

Step 1.2

Factor the perfect power ${(y^{2})}^{3}$ out of $y^{8}$ .

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

Step 1.3

Rearrange the fraction $\frac{1^{3} x}{{(y^{2})}^{3} y^{2}}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine.

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

Step 5

Multiply $\sqrt[3]{x}$ by $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $2$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 7.6.4

Multiply $2$ by $3$ .

$\frac{\sqrt[3]{x} {\sqrt[3]{y^{2}}}^{2}}{y^{2} y^{\frac{6}{3}}}$

Step 7.6.5

Cancel the common factor of $6$ and $3$ .

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Step 7.6.5.1

Factor $3$ out of $6$ .

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

Step 7.6.5.2

Cancel the common factors.

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Step 7.6.5.2.1

Factor $3$ out of $3$ .

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

Step 7.6.5.2.2

Cancel the common factor.

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

Step 7.6.5.2.3

Rewrite the expression.

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

Step 7.6.5.2.4

Divide $2$ by $1$ .

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

Step 8

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 8.1

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

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

Step 8.2

Add $2$ and $2$ .

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

Step 9

Simplify the numerator.

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

Multiply $2$ by $2$ .

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

Step 9.3

Factor out $y^{3}$ .

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

Step 9.4

Pull terms out from under the radical.

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

Step 9.5

Combine using the product rule for radicals.

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

Step 10

Cancel the common factor of $y$ and $y^{4}$ .

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Step 10.1

Factor $y$ out of $y \sqrt[3]{x y}$ .

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

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $y$ out of $y^{4}$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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