Simplify ( square root of a^9)/( fifth root of a^6)

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

Simplify the numerator.

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

Factor out $a^{8}$ .

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

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

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

Pull terms out from under the radical.

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

Simplify the denominator.

Factor out $a^{5}$ .

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

Pull terms out from under the radical.

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $4$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

Cancel the common factors.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Rewrite ${\sqrt[5]{a}}^{4}$ as ${(a^{4})}^{\frac{1}{5}}$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $8$ and $5$ .

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

Factor out $a^{10}$ .

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

Pull terms out from under the radical.

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

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

Move $a$ .

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

Multiply $a$ by $a^{2}$ .

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

$a^{1} a^{2} \sqrt[10]{a^{3}}$

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

$a^{1 + 2} \sqrt[10]{a^{3}}$

Add $1$ and $2$ .

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

Simplify ( square root of a^9)/( fifth root of a^6)

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