Algebra Examples

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

Step 1

Rewrite $\sqrt[5]{\frac{3}{4 y}}$ as $\frac{\sqrt[5]{3}}{\sqrt[5]{4 y}}$ .

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

Step 2

Multiply $\frac{\sqrt[5]{3}}{\sqrt[5]{4 y}}$ by $\frac{{\sqrt[5]{4 y}}^{4}}{{\sqrt[5]{4 y}}^{4}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[5]{3}}{\sqrt[5]{4 y}}$ by $\frac{{\sqrt[5]{4 y}}^{4}}{{\sqrt[5]{4 y}}^{4}}$ .

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

Step 3.2

Raise $\sqrt[5]{4 y}$ to the power of $1$ .

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{\sqrt[5]{4 y}}^{1} {\sqrt[5]{4 y}}^{4}}$

Step 3.3

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

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{\sqrt[5]{4 y}}^{1 + 4}}$

Step 3.4

Add $1$ and $4$ .

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

Step 3.5

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

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4 y}$ as ${(4 y)}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{({(4 y)}^{\frac{1}{5}})}^{5}}$

Step 3.5.2

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

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{(4 y)}^{\frac{1}{5} \cdot 5}}$

Step 3.5.3

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

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{(4 y)}^{\frac{5}{5}}}$

Step 3.5.4

Cancel the common factor of $5$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{(4 y)}^{\frac{5}{5}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[5]{3} {\sqrt[5]{4 y}}^{4}}{{(4 y)}^{1}}$

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{\sqrt[5]{3} \sqrt[5]{{(4 y)}^{4}}}{4 y}$

Step 4.2

Apply the product rule to $4 y$ .

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

Step 4.3

Raise $4$ to the power of $4$ .

$\frac{\sqrt[5]{3} \sqrt[5]{256 y^{4}}}{4 y}$

Step 4.4

Rewrite $256 y^{4}$ as $2^{5} \cdot (8 y^{4})$ .

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Step 4.4.1

Factor $32$ out of $256$ .

$\frac{\sqrt[5]{3} \sqrt[5]{32 (8) y^{4}}}{4 y}$

Step 4.4.2

Rewrite $32$ as $2^{5}$ .

$\frac{\sqrt[5]{3} \sqrt[5]{2^{5} \cdot 8 y^{4}}}{4 y}$

Step 4.4.3

Add parentheses.

$\frac{\sqrt[5]{3} \sqrt[5]{2^{5} \cdot (8 y^{4})}}{4 y}$

Step 4.5

Pull terms out from under the radical.

$\frac{\sqrt[5]{3} \cdot 2 \sqrt[5]{8 y^{4}}}{4 y}$

Step 4.6

Combine exponents.

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Step 4.6.1

Combine using the product rule for radicals.

$\frac{2 \sqrt[5]{3 (8 y^{4})}}{4 y}$

Step 4.6.2

Multiply $3$ by $8$ .

$\frac{2 \sqrt[5]{24 y^{4}}}{4 y}$

Step 5

Cancel the common factor of $2$ and $4$ .

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Step 5.1

Factor $2$ out of $2 \sqrt[5]{24 y^{4}}$ .

$\frac{2 (\sqrt[5]{24 y^{4}})}{4 y}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $4 y$ .

$\frac{2 (\sqrt[5]{24 y^{4}})}{2 (2 y)}$

Step 5.2.2

Cancel the common factor.

$\frac{2 \sqrt[5]{24 y^{4}}}{2 (2 y)}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[5]{24 y^{4}}}{2 y}$