Algebra Examples

$\frac{\sqrt{a^{9}}}{\sqrt[5]{a^{6}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $a^{9}$ as ${(a^{4})}^{2} a$ .

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Step 1.1.1

Factor out $a^{8}$ .

$\frac{\sqrt{a^{8} a}}{\sqrt[5]{a^{6}}}$

Step 1.1.2

Rewrite $a^{8}$ as ${(a^{4})}^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Remove non-negative terms from the absolute value.

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

Step 2

Simplify the denominator.

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Step 2.1

Factor out $a^{5}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor $a$ out of $a^{4} \sqrt{a}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $4$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $5$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 6

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

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Step 6.1

Factor $a$ out of $a^{3} \sqrt{a} {\sqrt[5]{a}}^{4}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Factor $a$ out of $a^{1}$ .

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.2.5

Divide $a^{2} \sqrt{a} {\sqrt[5]{a}}^{4}$ by $1$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

Rewrite the expression using the least common index of $10$ .

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Step 8.1.1

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

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

Step 8.1.2

Rewrite $a^{\frac{4}{5}}$ as $a^{\frac{8}{10}}$ .

$a^{2} (a^{\frac{8}{10}} \sqrt{a})$

Step 8.1.3

Rewrite $a^{\frac{8}{10}}$ as $\sqrt[10]{a^{8}}$ .

$a^{2} (\sqrt[10]{a^{8}} \sqrt{a})$

Step 8.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

$a^{2} (\sqrt[10]{a^{8}} a^{\frac{1}{2}})$

Step 8.1.5

Rewrite $a^{\frac{1}{2}}$ as $a^{\frac{5}{10}}$ .

$a^{2} (\sqrt[10]{a^{8}} a^{\frac{5}{10}})$

Step 8.1.6

Rewrite $a^{\frac{5}{10}}$ as $\sqrt[10]{a^{5}}$ .

$a^{2} (\sqrt[10]{a^{8}} \sqrt[10]{a^{5}})$

Step 8.2

Combine using the product rule for radicals.

$a^{2} \sqrt[10]{a^{8} a^{5}}$

Step 8.3

Multiply $a^{8}$ by $a^{5}$ by adding the exponents.

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Step 8.3.1

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

$a^{2} \sqrt[10]{a^{8 + 5}}$

Step 8.3.2

Add $8$ and $5$ .

$a^{2} \sqrt[10]{a^{13}}$

Step 9

Factor out $a^{10}$ .

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

Step 10

Pull terms out from under the radical.

$a^{2} | a | \sqrt[10]{a^{3}}$