Simplify ( cube root of m^6)( cube root of 32m^3)

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

Step 1

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

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

Step 2

Pull terms out from under the radical, assuming real numbers.

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

Step 3

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

Factor $8$ out of $32$ .

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

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

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

Move $4$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

Move $m$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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