Simplify sixth root of a^5 square root of a^3

$\sqrt[6]{a^{5}} \sqrt{a^{3}}$

Step 1

Factor out $a^{2}$ .

$\sqrt[6]{a^{5}} \sqrt{a^{2} a}$

Step 2

Pull terms out from under the radical.

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

Step 3

Multiply $\sqrt[6]{a^{5}} (a \sqrt{a})$ .

Rewrite the expression using the least common index of $6$ .

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

$a (\sqrt[6]{a^{5}} a^{\frac{1}{2}})$

Rewrite $a^{\frac{1}{2}}$ as $a^{\frac{3}{6}}$ .

$a (\sqrt[6]{a^{5}} a^{\frac{3}{6}})$

Rewrite $a^{\frac{3}{6}}$ as $\sqrt[6]{a^{3}}$ .

$a (\sqrt[6]{a^{5}} \sqrt[6]{a^{3}})$

Combine using the product rule for radicals.

$a \sqrt[6]{a^{5} a^{3}}$

Multiply $a^{5}$ by $a^{3}$ by adding the exponents.

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

$a \sqrt[6]{a^{5 + 3}}$

Add $5$ and $3$ .

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

Step 4

Factor out $a^{6}$ .

$a \sqrt[6]{a^{6} a^{2}}$

Step 5

Pull terms out from under the radical.

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

Step 6

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

Move $a$ .

$a \cdot a \sqrt[6]{a^{2}}$

Multiply $a$ by $a$ .

$a^{2} \sqrt[6]{a^{2}}$

Step 7

Rewrite $\sqrt[6]{a^{2}}$ as $\sqrt[3]{\sqrt{a^{2}}}$ .

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

Step 8

Pull terms out from under the radical, assuming positive real numbers.

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