Simplify cube root of 5v^4* cube root of 25v^6

$\sqrt[3]{5 v^{4}} \cdot \sqrt[3]{25 v^{6}}$

Rewrite $5 v^{4}$ as $v^{3} \cdot (5 v)$ .

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Factor out $v^{3}$ .

$\sqrt[3]{5 (v^{3} v)} \cdot \sqrt[3]{25 v^{6}}$

Reorder $5$ and $v^{3}$ .

$\sqrt[3]{v^{3} \cdot 5 v} \cdot \sqrt[3]{25 v^{6}}$

Add parentheses.

$\sqrt[3]{v^{3} \cdot (5 v)} \cdot \sqrt[3]{25 v^{6}}$

Pull terms out from under the radical.

$v \sqrt[3]{5 v} \cdot \sqrt[3]{25 v^{6}}$

Rewrite $25 v^{6}$ as ${(v^{2})}^{3} \cdot 25$ .

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Rewrite $v^{6}$ as ${(v^{2})}^{3}$ .

$v \sqrt[3]{5 v} \cdot \sqrt[3]{25 {(v^{2})}^{3}}$

Reorder $25$ and ${(v^{2})}^{3}$ .

$v \sqrt[3]{5 v} \cdot \sqrt[3]{{(v^{2})}^{3} \cdot 25}$

Pull terms out from under the radical.

$v \sqrt[3]{5 v} \cdot (v^{2} \sqrt[3]{25})$

Rewrite using the commutative property of multiplication.

$\sqrt[3]{25} (v \sqrt[3]{5 v}) v^{2}$

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

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Move $v^{2}$ .

$\sqrt[3]{25} (v^{2} v \sqrt[3]{5 v})$

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

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

$\sqrt[3]{25} (v^{2} v^{1} \sqrt[3]{5 v})$

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

$\sqrt[3]{25} (v^{2 + 1} \sqrt[3]{5 v})$

Add $2$ and $1$ .

$\sqrt[3]{25} (v^{3} \sqrt[3]{5 v})$

Multiply $\sqrt[3]{25} (v^{3} \sqrt[3]{5 v})$ .

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Combine using the product rule for radicals.

$v^{3} \sqrt[3]{25 (5 v)}$

Multiply $25$ by $5$ .

$v^{3} \sqrt[3]{125 v}$

Rewrite $125$ as $5^{3}$ .

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

Pull terms out from under the radical.

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

Rewrite using the commutative property of multiplication.

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

Simplify cube root of 5v^4* cube root of 25v^6

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