Simplify ( cube root of 2v^5)( cube root of 18v^6)

$(\sqrt[3]{2 v^{5}}) (\sqrt[3]{18 v^{6}})$

Step 1

Rewrite $2 v^{5}$ as $v^{3} \cdot (2 v^{2})$ .

Factor out $v^{3}$ .

$\sqrt[3]{2 (v^{3} v^{2})} \sqrt[3]{18 v^{6}}$

Reorder $2$ and $v^{3}$ .

$\sqrt[3]{v^{3} \cdot 2 v^{2}} \sqrt[3]{18 v^{6}}$

Add parentheses.

$\sqrt[3]{v^{3} \cdot (2 v^{2})} \sqrt[3]{18 v^{6}}$

Step 2

Pull terms out from under the radical.

$v \sqrt[3]{2 v^{2}} \sqrt[3]{18 v^{6}}$

Step 3

Rewrite $18 v^{6}$ as ${(v^{2})}^{3} \cdot 18$ .

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

$v \sqrt[3]{2 v^{2}} \sqrt[3]{18 {(v^{2})}^{3}}$

Reorder $18$ and ${(v^{2})}^{3}$ .

$v \sqrt[3]{2 v^{2}} \sqrt[3]{{(v^{2})}^{3} \cdot 18}$

Step 4

Pull terms out from under the radical.

$v \sqrt[3]{2 v^{2}} (v^{2} \sqrt[3]{18})$

Step 5

Rewrite using the commutative property of multiplication.

$\sqrt[3]{18} v \sqrt[3]{2 v^{2}} v^{2}$

Step 6

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

Move $v^{2}$ .

$\sqrt[3]{18} (v^{2} v) \sqrt[3]{2 v^{2}}$

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

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

$\sqrt[3]{18} (v^{2} v^{1}) \sqrt[3]{2 v^{2}}$

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

$\sqrt[3]{18} v^{2 + 1} \sqrt[3]{2 v^{2}}$

Add $2$ and $1$ .

$\sqrt[3]{18} v^{3} \sqrt[3]{2 v^{2}}$

Step 7

Multiply $\sqrt[3]{18} v^{3} \sqrt[3]{2 v^{2}}$ .

Combine using the product rule for radicals.

$v^{3} \sqrt[3]{18 (2 v^{2})}$

Multiply $18$ by $2$ .

$v^{3} \sqrt[3]{36 v^{2}}$