Algebra Examples

$\frac{\sqrt[3]{x^{6}}}{\sqrt[5]{x^{6}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\sqrt[3]{{(x^{2})}^{3}}}{\sqrt[5]{x^{6}}}$

Step 1.2

Pull terms out from under the radical, assuming real numbers.

$\frac{x^{2}}{\sqrt[5]{x^{6}}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor out $x^{5}$ .

$\frac{x^{2}}{\sqrt[5]{x^{5} x}}$

Step 2.2

Pull terms out from under the radical.

$\frac{x^{2}}{x \sqrt[5]{x}}$

Step 3

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

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Step 3.1

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

$\frac{x \cdot x}{x \sqrt[5]{x}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x$ out of $x \sqrt[5]{x}$ .

$\frac{x \cdot x}{x (\sqrt[5]{x})}$

Step 3.2.2

Cancel the common factor.

$\frac{x \cdot x}{x \sqrt[5]{x}}$

Step 3.2.3

Rewrite the expression.

$\frac{x}{\sqrt[5]{x}}$

Step 4

Multiply $\frac{x}{\sqrt[5]{x}}$ by $\frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$ .

$\frac{x}{\sqrt[5]{x}} \cdot \frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{x}{\sqrt[5]{x}}$ by $\frac{{\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{4}}$ .

$\frac{x {\sqrt[5]{x}}^{4}}{\sqrt[5]{x} {\sqrt[5]{x}}^{4}}$

Step 5.2

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

$\frac{x {\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{1} {\sqrt[5]{x}}^{4}}$

Step 5.3

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

$\frac{x {\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{1 + 4}}$

Step 5.4

Add $1$ and $4$ .

$\frac{x {\sqrt[5]{x}}^{4}}{{\sqrt[5]{x}}^{5}}$

Step 5.5

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

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Step 5.5.1

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

$\frac{x {\sqrt[5]{x}}^{4}}{{(x^{\frac{1}{5}})}^{5}}$

Step 5.5.2

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

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

Step 5.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{x {\sqrt[5]{x}}^{4}}{x^{\frac{5}{5}}}$

Step 5.5.4

Cancel the common factor of $5$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{x {\sqrt[5]{x}}^{4}}{x^{\frac{5}{5}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{x {\sqrt[5]{x}}^{4}}{x^{1}}$

Step 5.5.5

Simplify.

$\frac{x {\sqrt[5]{x}}^{4}}{x}$

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Cancel the common factor.

$\frac{x {\sqrt[5]{x}}^{4}}{x}$

Step 6.2

Divide ${\sqrt[5]{x}}^{4}$ by $1$ .

${\sqrt[5]{x}}^{4}$

Step 7

Rewrite ${\sqrt[5]{x}}^{4}$ as $\sqrt[5]{x^{4}}$ .

$\sqrt[5]{x^{4}}$