Algebra Examples

$\frac{\sqrt[5]{a^{11}}}{\sqrt{a}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $a^{11}$ as ${(a^{2})}^{5} a$ .

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Step 1.1.1

Factor out $a^{10}$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Rewrite ${\sqrt{a}}^{2}$ as $a$ .

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

$\frac{a \sqrt[5]{a} \sqrt{a}}{1}$

Step 4.2.5

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

$a \sqrt[5]{a} \sqrt{a}$

Step 5

Multiply $a \sqrt[5]{a} \sqrt{a}$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

$a (\sqrt[10]{a^{5}} a^{\frac{1}{5}})$

Step 5.1.5

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

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

Step 5.1.6

Rewrite $a^{\frac{2}{10}}$ as $\sqrt[10]{a^{2}}$ .

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

Multiply $a^{5}$ by $a^{2}$ by adding the exponents.

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Step 5.3.1

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

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

Step 5.3.2

Add $5$ and $2$ .

$a \sqrt[10]{a^{7}}$