Algebra Examples

$\sqrt[4]{\frac{5 x}{8 y}}$

Step 1

Rewrite $\sqrt[4]{\frac{5 x}{8 y}}$ as $\frac{\sqrt[4]{5 x}}{\sqrt[4]{8 y}}$ .

$\frac{\sqrt[4]{5 x}}{\sqrt[4]{8 y}}$

Step 2

Multiply $\frac{\sqrt[4]{5 x}}{\sqrt[4]{8 y}}$ by $\frac{{\sqrt[4]{8 y}}^{3}}{{\sqrt[4]{8 y}}^{3}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[4]{5 x}}{\sqrt[4]{8 y}}$ by $\frac{{\sqrt[4]{8 y}}^{3}}{{\sqrt[4]{8 y}}^{3}}$ .

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

Step 3.2

Raise $\sqrt[4]{8 y}$ to the power of $1$ .

$\frac{\sqrt[4]{5 x} {\sqrt[4]{8 y}}^{3}}{{\sqrt[4]{8 y}}^{1} {\sqrt[4]{8 y}}^{3}}$

Step 3.3

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

$\frac{\sqrt[4]{5 x} {\sqrt[4]{8 y}}^{3}}{{\sqrt[4]{8 y}}^{1 + 3}}$

Step 3.4

Add $1$ and $3$ .

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

Step 3.5

Rewrite ${\sqrt[4]{8 y}}^{4}$ as $8 y$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8 y}$ as ${(8 y)}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{5 x} {\sqrt[4]{8 y}}^{3}}{{({(8 y)}^{\frac{1}{4}})}^{4}}$

Step 3.5.2

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

$\frac{\sqrt[4]{5 x} {\sqrt[4]{8 y}}^{3}}{{(8 y)}^{\frac{1}{4} \cdot 4}}$

Step 3.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[4]{5 x} {\sqrt[4]{8 y}}^{3}}{{(8 y)}^{1}}$

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[4]{8 y}}^{3}$ as $\sqrt[4]{{(8 y)}^{3}}$ .

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

Step 4.2

Apply the product rule to $8 y$ .

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

Step 4.3

Raise $8$ to the power of $3$ .

$\frac{\sqrt[4]{5 x} \sqrt[4]{512 y^{3}}}{8 y}$

Step 4.4

Rewrite $512 y^{3}$ as $4^{4} \cdot (2 y^{3})$ .

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Step 4.4.1

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{5 x} \sqrt[4]{256 (2) y^{3}}}{8 y}$

Step 4.4.2

Rewrite $256$ as $4^{4}$ .

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

Step 4.4.3

Add parentheses.

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

Step 4.5

Pull terms out from under the radical.

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

Step 4.6

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

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

Step 4.7

Combine exponents.

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Step 4.7.1

Combine using the product rule for radicals.

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

Step 4.7.2

Multiply $2$ by $5$ .

$\frac{4 \sqrt[4]{10 x y^{3}}}{8 y}$

Step 5

Cancel the common factor of $4$ and $8$ .

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Step 5.1

Factor $4$ out of $4 \sqrt[4]{10 x y^{3}}$ .

$\frac{4 (\sqrt[4]{10 x y^{3}})}{8 y}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $4$ out of $8 y$ .

$\frac{4 (\sqrt[4]{10 x y^{3}})}{4 (2 y)}$

Step 5.2.2

Cancel the common factor.

$\frac{4 \sqrt[4]{10 x y^{3}}}{4 (2 y)}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[4]{10 x y^{3}}}{2 y}$