Algebra Examples

$\sqrt[3]{\frac{x^{6}}{x^{10}}}$

Step 1

Reduce the expression $\frac{x^{6}}{x^{10}}$ by cancelling the common factors.

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Step 1.1

Multiply by $1$ .

$\sqrt[3]{\frac{x^{6} \cdot 1}{x^{10}}}$

Step 1.2

Factor $x^{6}$ out of $x^{10}$ .

$\sqrt[3]{\frac{x^{6} \cdot 1}{x^{6} x^{4}}}$

Step 1.3

Cancel the common factor.

$\sqrt[3]{\frac{x^{6} \cdot 1}{x^{6} x^{4}}}$

Step 1.4

Rewrite the expression.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Rearrange the fraction $\frac{1^{3} \cdot 1}{x^{3} x}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

Rewrite $\sqrt[3]{\frac{1}{x}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{x}}$ .

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

Step 5

Any root of $1$ is $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $2$ .

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

Step 7.5

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

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

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

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

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

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

Step 7.5.4.2

Rewrite the expression.

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

Step 7.5.5

Simplify.

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

Step 8

Combine.

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

Step 9

Simplify the expression.

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Step 9.1

Multiply $x$ by $x$ .

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

Step 9.2

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

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

Step 10

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

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