Simplify fifth root of a^9 square root of a

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

Factor out $a^{5}$ .

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

Pull terms out from under the radical.

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

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

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

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

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

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

$a (a^{\frac{5}{10}} \sqrt[5]{a^{4}})$

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

$a (\sqrt[10]{a^{5}} \sqrt[5]{a^{4}})$

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

$a (\sqrt[10]{a^{5}} a^{\frac{4}{5}})$

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

Add $5$ and $8$ .

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

Factor out $a^{10}$ .

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

Pull terms out from under the radical.

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

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

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

Multiply $a$ by $a$ .

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