Algebra Examples

$\sqrt[5]{\frac{a^{2}}{x^{3}}}$

Step 1

Rewrite $\sqrt[5]{\frac{a^{2}}{x^{3}}}$ as $\frac{\sqrt[5]{a^{2}}}{\sqrt[5]{x^{3}}}$ .

$\frac{\sqrt[5]{a^{2}}}{\sqrt[5]{x^{3}}}$

Step 2

Multiply $\frac{\sqrt[5]{a^{2}}}{\sqrt[5]{x^{3}}}$ by $\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{4}}$ .

$\frac{\sqrt[5]{a^{2}}}{\sqrt[5]{x^{3}}} \cdot \frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{4}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[5]{a^{2}}}{\sqrt[5]{x^{3}}}$ by $\frac{{\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{4}}$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{\sqrt[5]{x^{3}} {\sqrt[5]{x^{3}}}^{4}}$

Step 3.2

Raise $\sqrt[5]{x^{3}}$ to the power of $1$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{1} {\sqrt[5]{x^{3}}}^{4}}$

Step 3.3

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

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{1 + 4}}$

Step 3.4

Add $1$ and $4$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{{\sqrt[5]{x^{3}}}^{5}}$

Step 3.5

Rewrite ${\sqrt[5]{x^{3}}}^{5}$ as $x^{3}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{{(x^{\frac{3}{5}})}^{5}}$

Step 3.5.2

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

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{3}{5} \cdot 5}}$

Step 3.5.3

Combine $\frac{3}{5}$ and $5$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{3 \cdot 5}{5}}}$

Step 3.5.4

Multiply $3$ by $5$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{15}{5}}}$

Step 3.5.5

Cancel the common factor of $15$ and $5$ .

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Step 3.5.5.1

Factor $5$ out of $15$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{5 \cdot 3}{5}}}$

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $5$ out of $5$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{5 \cdot 3}{5 (1)}}}$

Step 3.5.5.2.2

Cancel the common factor.

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{5 \cdot 3}{5 \cdot 1}}}$

Step 3.5.5.2.3

Rewrite the expression.

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{\frac{3}{1}}}$

Step 3.5.5.2.4

Divide $3$ by $1$ .

$\frac{\sqrt[5]{a^{2}} {\sqrt[5]{x^{3}}}^{4}}{x^{3}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[5]{x^{3}}}^{4}$ as $\sqrt[5]{{(x^{3})}^{4}}$ .

$\frac{\sqrt[5]{a^{2}} \sqrt[5]{{(x^{3})}^{4}}}{x^{3}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{\sqrt[5]{a^{2}} \sqrt[5]{x^{3 \cdot 4}}}{x^{3}}$

Step 4.2.2

Multiply $3$ by $4$ .

$\frac{\sqrt[5]{a^{2}} \sqrt[5]{x^{12}}}{x^{3}}$

Step 4.3

Rewrite $x^{12}$ as ${(x^{2})}^{5} x^{2}$ .

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Step 4.3.1

Factor out $x^{10}$ .

$\frac{\sqrt[5]{a^{2}} \sqrt[5]{x^{10} x^{2}}}{x^{3}}$

Step 4.3.2

Rewrite $x^{10}$ as ${(x^{2})}^{5}$ .

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

Step 4.4

Pull terms out from under the radical.

$\frac{\sqrt[5]{a^{2}} x^{2} \sqrt[5]{x^{2}}}{x^{3}}$

Step 4.5

Combine using the product rule for radicals.

$\frac{x^{2} \sqrt[5]{a^{2} x^{2}}}{x^{3}}$

Step 5

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.1

Factor $x^{2}$ out of $x^{2} \sqrt[5]{a^{2} x^{2}}$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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