Simplify ( cube root of 3q^7)( cube root of 32q^3)

$(\sqrt[3]{3 q^{7}}) (\sqrt[3]{32 q^{3}})$

Rewrite $3 q^{7}$ as ${(q^{2})}^{3} \cdot (3 q)$ .

Factor out $q^{6}$ .

$\sqrt[3]{3 (q^{6} q)} \sqrt[3]{32 q^{3}}$

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

$\sqrt[3]{3 ({(q^{2})}^{3} q)} \sqrt[3]{32 q^{3}}$

Reorder $3$ and ${(q^{2})}^{3}$ .

$\sqrt[3]{{(q^{2})}^{3} \cdot 3 q} \sqrt[3]{32 q^{3}}$

Add parentheses.

$\sqrt[3]{{(q^{2})}^{3} \cdot (3 q)} \sqrt[3]{32 q^{3}}$

Pull terms out from under the radical.

$q^{2} \sqrt[3]{3 q} \sqrt[3]{32 q^{3}}$

Rewrite $32 q^{3}$ as ${(2 q)}^{3} \cdot 4$ .

Factor $8$ out of $32$ .

$q^{2} \sqrt[3]{3 q} \sqrt[3]{8 (4) q^{3}}$

Rewrite $8$ as $2^{3}$ .

$q^{2} \sqrt[3]{3 q} \sqrt[3]{2^{3} \cdot 4 q^{3}}$

Move $4$ .

$q^{2} \sqrt[3]{3 q} \sqrt[3]{2^{3} q^{3} \cdot 4}$

Rewrite $2^{3} q^{3}$ as ${(2 q)}^{3}$ .

$q^{2} \sqrt[3]{3 q} \sqrt[3]{{(2 q)}^{3} \cdot 4}$

Pull terms out from under the radical.

$q^{2} \sqrt[3]{3 q} (2 q \sqrt[3]{4})$

Rewrite using the commutative property of multiplication.

$2 \sqrt[3]{4} q^{2} \sqrt[3]{3 q} q$

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

Move $q$ .

$2 \sqrt[3]{4} (q \cdot q^{2}) \sqrt[3]{3 q}$

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

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

$2 \sqrt[3]{4} (q^{1} q^{2}) \sqrt[3]{3 q}$

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

$2 \sqrt[3]{4} q^{1 + 2} \sqrt[3]{3 q}$

Add $1$ and $2$ .

$2 \sqrt[3]{4} q^{3} \sqrt[3]{3 q}$

Multiply $2 \sqrt[3]{4} q^{3} \sqrt[3]{3 q}$ .

Combine using the product rule for radicals.

$2 q^{3} \sqrt[3]{4 (3 q)}$

Multiply $4$ by $3$ .

$2 q^{3} \sqrt[3]{12 q}$