Algebra Examples

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

Step 1

Rewrite $\sqrt[3]{\frac{a}{b^{2}}}$ as $\frac{\sqrt[3]{a}}{\sqrt[3]{b^{2}}}$ .

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

Raise $\sqrt[3]{b^{2}}$ to the power of $1$ .

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

Rewrite ${\sqrt[3]{b^{2}}}^{3}$ as $b^{2}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{b^{2}}$ as $b^{\frac{2}{3}}$ .

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

Step 3.5.2

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

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

Step 3.5.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 3.5.4

Multiply $2$ by $3$ .

$\frac{\sqrt[3]{a} {\sqrt[3]{b^{2}}}^{2}}{b^{\frac{6}{3}}}$

Step 3.5.5

Cancel the common factor of $6$ and $3$ .

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Step 3.5.5.1

Factor $3$ out of $6$ .

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

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $3$ out of $3$ .

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

Step 3.5.5.2.2

Cancel the common factor.

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

Step 3.5.5.2.3

Rewrite the expression.

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

Step 3.5.5.2.4

Divide $2$ by $1$ .

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Multiply the exponents in ${(b^{2})}^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt[3]{a} \sqrt[3]{b^{4}}}{b^{2}}$

Step 4.3

Factor out $b^{3}$ .

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

Step 4.4

Pull terms out from under the radical.

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

Step 4.5

Combine using the product rule for radicals.

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

Step 5

Cancel the common factor of $b$ and $b^{2}$ .

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Step 5.1

Factor $b$ out of $b \sqrt[3]{a b}$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $b$ out of $b^{2}$ .

$\frac{b (\sqrt[3]{a b})}{b \cdot b}$

Step 5.2.2

Cancel the common factor.

$\frac{b \sqrt[3]{a b}}{b \cdot b}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[3]{a b}}{b}$