Pre-Algebra Examples

$\sqrt[4]{\frac{5 x}{243 y^{16}}}$

Step 1

Rewrite $\frac{5 x}{243 y^{16}}$ as ${(\frac{1}{3 y^{4}})}^{4} \frac{5 x}{3}$ .

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Step 1.1

Factor the perfect power $1^{4}$ out of $5 x$ .

$\sqrt[4]{\frac{1^{4} (5 x)}{243 y^{16}}}$

Step 1.2

Factor the perfect power ${(3 y^{4})}^{4}$ out of $243 y^{16}$ .

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

Step 1.3

Rearrange the fraction $\frac{1^{4} (5 x)}{{(3 y^{4})}^{4} \cdot 3}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine.

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

Step 5

Multiply $\sqrt[4]{5 x}$ by $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

Move $\sqrt[4]{3}$ .

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

Step 7.3

Raise $\sqrt[4]{3}$ to the power of $1$ .

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

Step 7.4

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

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

Step 7.5

Add $1$ and $3$ .

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

Step 7.6

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $4$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

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

Step 8.2

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

$\frac{\sqrt[4]{5 x} \sqrt[4]{27}}{3 y^{4} \cdot 3}$

Step 9

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{5 x} \sqrt[4]{27}}{9 y^{4}}$

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

$\frac{\sqrt[4]{5 x \cdot 27}}{9 y^{4}}$

Step 10.2

Multiply $27$ by $5$ .

$\frac{\sqrt[4]{135 x}}{9 y^{4}}$