Simplify fourth root of v fifth root of v^8

$\sqrt[4]{v} \sqrt[5]{v^{8}}$

Factor out $v^{5}$ .

$\sqrt[4]{v} \sqrt[5]{v^{5} v^{3}}$

Pull terms out from under the radical.

$\sqrt[4]{v} (v \sqrt[5]{v^{3}})$

Multiply $\sqrt[4]{v} (v \sqrt[5]{v^{3}})$ .

Rewrite the expression using the least common index of $20$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{v}$ as $v^{\frac{1}{4}}$ .

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

Rewrite $v^{\frac{1}{4}}$ as $v^{\frac{5}{20}}$ .

$v (v^{\frac{5}{20}} \sqrt[5]{v^{3}})$

Rewrite $v^{\frac{5}{20}}$ as $\sqrt[20]{v^{5}}$ .

$v (\sqrt[20]{v^{5}} \sqrt[5]{v^{3}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{v^{3}}$ as $v^{\frac{3}{5}}$ .

$v (\sqrt[20]{v^{5}} v^{\frac{3}{5}})$

Rewrite $v^{\frac{3}{5}}$ as $v^{\frac{12}{20}}$ .

$v (\sqrt[20]{v^{5}} v^{\frac{12}{20}})$

Rewrite $v^{\frac{12}{20}}$ as $\sqrt[20]{v^{12}}$ .

$v (\sqrt[20]{v^{5}} \sqrt[20]{v^{12}})$

Combine using the product rule for radicals.

$v \sqrt[20]{v^{5} v^{12}}$

Multiply $v^{5}$ by $v^{12}$ by adding the exponents.

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

$v \sqrt[20]{v^{5 + 12}}$

Add $5$ and $12$ .

$v \sqrt[20]{v^{17}}$