Algebra Examples

$\sqrt[5]{\frac{32 x^{6}}{y^{11}}}$

Step 1

Rewrite $\frac{32 x^{6}}{y^{11}}$ as ${(\frac{2 x}{y^{2}})}^{5} \frac{x}{y}$ .

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Step 1.1

Factor the perfect power ${(2 x)}^{5}$ out of $32 x^{6}$ .

$\sqrt[5]{\frac{{(2 x)}^{5} x}{y^{11}}}$

Step 1.2

Factor the perfect power ${(y^{2})}^{5}$ out of $y^{11}$ .

$\sqrt[5]{\frac{{(2 x)}^{5} x}{{(y^{2})}^{5} y}}$

Step 1.3

Rearrange the fraction $\frac{{(2 x)}^{5} x}{{(y^{2})}^{5} y}$ .

$\sqrt[5]{{(\frac{2 x}{y^{2}})}^{5} \frac{x}{y}}$

Step 2

Pull terms out from under the radical.

$\frac{2 x}{y^{2}} \sqrt[5]{\frac{x}{y}}$

Step 3

Rewrite $\sqrt[5]{\frac{x}{y}}$ as $\frac{\sqrt[5]{x}}{\sqrt[5]{y}}$ .

$\frac{2 x}{y^{2}} \cdot \frac{\sqrt[5]{x}}{\sqrt[5]{y}}$

Step 4

Combine.

$\frac{2 x \sqrt[5]{x}}{y^{2} \sqrt[5]{y}}$

Step 5

Multiply $\frac{2 x \sqrt[5]{x}}{y^{2} \sqrt[5]{y}}$ by $\frac{{\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{4}}$ .

$\frac{2 x \sqrt[5]{x}}{y^{2} \sqrt[5]{y}} \cdot \frac{{\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{4}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{2 x \sqrt[5]{x}}{y^{2} \sqrt[5]{y}}$ by $\frac{{\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{4}}$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} \sqrt[5]{y} {\sqrt[5]{y}}^{4}}$

Step 6.2

Move $\sqrt[5]{y}$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} (\sqrt[5]{y} {\sqrt[5]{y}}^{4})}$

Step 6.3

Raise $\sqrt[5]{y}$ to the power of $1$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} ({\sqrt[5]{y}}^{1} {\sqrt[5]{y}}^{4})}$

Step 6.4

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

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} {\sqrt[5]{y}}^{1 + 4}}$

Step 6.5

Add $1$ and $4$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} {\sqrt[5]{y}}^{5}}$

Step 6.6

Rewrite ${\sqrt[5]{y}}^{5}$ as $y$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y}$ as $y^{\frac{1}{5}}$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} {(y^{\frac{1}{5}})}^{5}}$

Step 6.6.2

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

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} y^{\frac{1}{5} \cdot 5}}$

Step 6.6.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} y^{\frac{5}{5}}}$

Step 6.6.4

Cancel the common factor of $5$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} y^{\frac{5}{5}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} y^{1}}$

Step 6.6.5

Simplify.

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} y}$

Step 7

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

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Step 7.1

Multiply $y^{2}$ by $y$ .

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Step 7.1.1

Raise $y$ to the power of $1$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2} y^{1}}$

Step 7.1.2

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

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{2 + 1}}$

Step 7.2

Add $2$ and $1$ .

$\frac{2 x \sqrt[5]{x} {\sqrt[5]{y}}^{4}}{y^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[5]{y}}^{4}$ as $\sqrt[5]{y^{4}}$ .

$\frac{2 x \sqrt[5]{x} \sqrt[5]{y^{4}}}{y^{3}}$

Step 8.2

Combine using the product rule for radicals.

$\frac{2 x \sqrt[5]{y^{4} x}}{y^{3}}$