Algebra Examples

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

Step 1

Rewrite $\frac{3}{5 a^{4}}$ as ${(\frac{1}{a})}^{3} \frac{3}{5 a}$ .

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Step 1.1

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

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

Step 1.2

Factor the perfect power $a^{3}$ out of $5 a^{4}$ .

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

Step 1.3

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

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

Step 2

Pull terms out from under the radical.

$\frac{1}{a} \sqrt[3]{\frac{3}{5 a}}$

Step 3

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

$\frac{1}{a} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{5 a}}$

Step 4

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{5 a}}$ by $\frac{{\sqrt[3]{5 a}}^{2}}{{\sqrt[3]{5 a}}^{2}}$ .

$\frac{1}{a} (\frac{\sqrt[3]{3}}{\sqrt[3]{5 a}} \cdot \frac{{\sqrt[3]{5 a}}^{2}}{{\sqrt[3]{5 a}}^{2}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{5 a}}$ by $\frac{{\sqrt[3]{5 a}}^{2}}{{\sqrt[3]{5 a}}^{2}}$ .

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{\sqrt[3]{5 a} {\sqrt[3]{5 a}}^{2}}$

Step 5.2

Raise $\sqrt[3]{5 a}$ to the power of $1$ .

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{\sqrt[3]{5 a}}^{1} {\sqrt[3]{5 a}}^{2}}$

Step 5.3

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

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{\sqrt[3]{5 a}}^{1 + 2}}$

Step 5.4

Add $1$ and $2$ .

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{\sqrt[3]{5 a}}^{3}}$

Step 5.5

Rewrite ${\sqrt[3]{5 a}}^{3}$ as $5 a$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 a}$ as ${(5 a)}^{\frac{1}{3}}$ .

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{({(5 a)}^{\frac{1}{3}})}^{3}}$

Step 5.5.2

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

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{(5 a)}^{\frac{1}{3} \cdot 3}}$

Step 5.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{(5 a)}^{\frac{3}{3}}}$

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{(5 a)}^{\frac{3}{3}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{{(5 a)}^{1}}$

Step 5.5.5

Simplify.

$\frac{1}{a} \cdot \frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{5 a}$

Step 6

Combine.

$\frac{1 (\sqrt[3]{3} {\sqrt[3]{5 a}}^{2})}{a (5 a)}$

Step 7

Multiply $a$ by $a$ by adding the exponents.

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Step 7.1

Move $a$ .

$\frac{1 (\sqrt[3]{3} {\sqrt[3]{5 a}}^{2})}{a \cdot a \cdot 5}$

Step 7.2

Multiply $a$ by $a$ .

$\frac{1 (\sqrt[3]{3} {\sqrt[3]{5 a}}^{2})}{a^{2} \cdot 5}$

Step 8

Multiply $\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}$ by $1$ .

$\frac{\sqrt[3]{3} {\sqrt[3]{5 a}}^{2}}{a^{2} \cdot 5}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite ${\sqrt[3]{5 a}}^{2}$ as $\sqrt[3]{{(5 a)}^{2}}$ .

$\frac{\sqrt[3]{3} \sqrt[3]{{(5 a)}^{2}}}{a^{2} \cdot 5}$

Step 9.2

Apply the product rule to $5 a$ .

$\frac{\sqrt[3]{3} \sqrt[3]{5^{2} a^{2}}}{a^{2} \cdot 5}$

Step 9.3

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{3} \sqrt[3]{25 a^{2}}}{a^{2} \cdot 5}$

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 (25 a^{2})}}{a^{2} \cdot 5}$

Step 10.2

Multiply $3$ by $25$ .

$\frac{\sqrt[3]{75 a^{2}}}{a^{2} \cdot 5}$

Step 11

Move $5$ to the left of $a^{2}$ .

$\frac{\sqrt[3]{75 a^{2}}}{5 a^{2}}$