Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{8}$ as ${(x^{2})}^{3} x^{2}$ .

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Step 1.1.1

Factor out $x^{6}$ .

$\frac{\sqrt[3]{x^{6} x^{2}}}{{(x^{4})}^{\frac{1}{3}}}$

Step 1.1.2

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{\sqrt[3]{{(x^{2})}^{3} x^{2}}}{{(x^{4})}^{\frac{1}{3}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[3]{x^{2}}}{{(x^{4})}^{\frac{1}{3}}}$

Step 2

Multiply the exponents in ${(x^{4})}^{\frac{1}{3}}$ .

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Step 2.1

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

$\frac{x^{2} \sqrt[3]{x^{2}}}{x^{4 (\frac{1}{3})}}$

Step 2.2

Combine $4$ and $\frac{1}{3}$ .

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

Step 3

Move $x^{\frac{4}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 4

Multiply $x^{2}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 4.1

Move $x^{- \frac{4}{3}}$ .

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

Step 4.2

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

$x^{- \frac{4}{3} + 2} \sqrt[3]{x^{2}}$

Step 4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{- \frac{4}{3} + 2 \cdot \frac{3}{3}} \sqrt[3]{x^{2}}$

Step 4.4

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

$x^{- \frac{4}{3} + \frac{2 \cdot 3}{3}} \sqrt[3]{x^{2}}$

Step 4.5

Combine the numerators over the common denominator.

$x^{\frac{- 4 + 2 \cdot 3}{3}} \sqrt[3]{x^{2}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $3$ .

$x^{\frac{- 4 + 6}{3}} \sqrt[3]{x^{2}}$

Step 4.6.2

Add $- 4$ and $6$ .

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

Step 5

Multiply $x^{\frac{2}{3}} \sqrt[3]{x^{2}}$ .

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Step 5.1

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

$x^{\frac{2}{3}} x^{\frac{2}{3}}$

Step 5.2

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

$x^{\frac{2}{3} + \frac{2}{3}}$

Step 5.3

Combine the numerators over the common denominator.

$x^{\frac{2 + 2}{3}}$

Step 5.4

Add $2$ and $2$ .

$x^{\frac{4}{3}}$