Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Multiply $2$ by $3$ .

$\frac{\sqrt[5]{a^{4}} {\sqrt[3]{a^{2}}}^{2}}{a^{\frac{6}{3}}}$

Step 2.5.5

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

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Step 2.5.5.1

Factor $3$ out of $6$ .

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

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $3$ out of $3$ .

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

Step 2.5.5.2.2

Cancel the common factor.

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

Step 2.5.5.2.3

Rewrite the expression.

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

Step 2.5.5.2.4

Divide $2$ by $1$ .

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.3

Factor out $a^{3}$ .

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

Step 3.4

Pull terms out from under the radical.

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

Step 3.5

Combine exponents.

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Step 3.5.1

Rewrite the expression using the least common index of $15$ .

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Step 3.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{a^{4}}$ as $a^{\frac{4}{5}}$ .

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

Step 3.5.1.2

Rewrite $a^{\frac{4}{5}}$ as $a^{\frac{12}{15}}$ .

$\frac{a (a^{\frac{12}{15}} \sqrt[3]{a})}{a^{2}}$

Step 3.5.1.3

Rewrite $a^{\frac{12}{15}}$ as $\sqrt[15]{a^{12}}$ .

$\frac{a (\sqrt[15]{a^{12}} \sqrt[3]{a})}{a^{2}}$

Step 3.5.1.4

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

$\frac{a (\sqrt[15]{a^{12}} a^{\frac{1}{3}})}{a^{2}}$

Step 3.5.1.5

Rewrite $a^{\frac{1}{3}}$ as $a^{\frac{5}{15}}$ .

$\frac{a (\sqrt[15]{a^{12}} a^{\frac{5}{15}})}{a^{2}}$

Step 3.5.1.6

Rewrite $a^{\frac{5}{15}}$ as $\sqrt[15]{a^{5}}$ .

$\frac{a (\sqrt[15]{a^{12}} \sqrt[15]{a^{5}})}{a^{2}}$

Step 3.5.2

Combine using the product rule for radicals.

$\frac{a \sqrt[15]{a^{12} a^{5}}}{a^{2}}$

Step 3.5.3

Multiply $a^{12}$ by $a^{5}$ by adding the exponents.

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Step 3.5.3.1

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

$\frac{a \sqrt[15]{a^{12 + 5}}}{a^{2}}$

Step 3.5.3.2

Add $12$ and $5$ .

$\frac{a \sqrt[15]{a^{17}}}{a^{2}}$

Step 3.6

Factor out $a^{15}$ .

$\frac{a \sqrt[15]{a^{15} a^{2}}}{a^{2}}$

Step 3.7

Pull terms out from under the radical.

$\frac{a (a \sqrt[15]{a^{2}})}{a^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Raise $a$ to the power of $1$ .

$\frac{a^{1} a \sqrt[15]{a^{2}}}{a^{2}}$

Step 4.2

Raise $a$ to the power of $1$ .

$\frac{a^{1} a^{1} \sqrt[15]{a^{2}}}{a^{2}}$

Step 4.3

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

$\frac{a^{1 + 1} \sqrt[15]{a^{2}}}{a^{2}}$

Step 4.4

Add $1$ and $1$ .

$\frac{a^{2} \sqrt[15]{a^{2}}}{a^{2}}$

Step 5

Cancel the common factor of $a^{2}$ .

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Step 5.1

Cancel the common factor.

$\frac{a^{2} \sqrt[15]{a^{2}}}{a^{2}}$

Step 5.2

Divide $\sqrt[15]{a^{2}}$ by $1$ .

$\sqrt[15]{a^{2}}$