Algebra Examples

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

Step 1

Factor out $v^{5}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.2

Combine using the product rule for radicals.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Add $5$ and $12$ .

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