Calculus Examples

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

Step 1

Simplify the denominator.

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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 $x^{6}$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{1}{x^{2} \sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1}{x^{2} \sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $2$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Multiply $2$ by $3$ .

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

Step 3.6.5

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

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Step 3.6.5.1

Factor $3$ out of $6$ .

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

Step 3.6.5.2

Cancel the common factors.

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Step 3.6.5.2.1

Factor $3$ out of $3$ .

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

Step 3.6.5.2.2

Cancel the common factor.

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

Step 3.6.5.2.3

Rewrite the expression.

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

Step 3.6.5.2.4

Divide $2$ by $1$ .

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

Step 4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 4.1

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

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

Step 4.2

Add $2$ and $2$ .

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $2$ .

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

Step 5.3

Factor out $x^{3}$ .

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

Step 5.4

Pull terms out from under the radical.

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

Step 6

Cancel the common factor of $x$ and $x^{4}$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $x$ out of $x^{4}$ .

$\frac{x (\sqrt[3]{x})}{x \cdot x^{3}}$

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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