Simplify the Radical Expression ( fifth root of a^11)/( square root of a)

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

Step 1

Simplify the numerator.

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

Factor out $a^{10}$ .

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

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

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

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.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

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

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

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

Cancel the common factors.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $5$ and $2$ .

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

Simplify the Radical Expression ( fifth root of a^11)/( square root of a)

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