Simplify ( cube root of 16a^5)( cube root of 2a^6)

$(\sqrt[3]{16 a^{5}}) (\sqrt[3]{2 a^{6}})$

Step 1

Rewrite $16 a^{5}$ as ${(2 a)}^{3} \cdot (2 a^{2})$ .

Factor $8$ out of $16$ .

$\sqrt[3]{8 (2) a^{5}} \sqrt[3]{2 a^{6}}$

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

$\sqrt[3]{2^{3} \cdot 2 a^{5}} \sqrt[3]{2 a^{6}}$

Factor out $a^{3}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$2 a \sqrt[3]{2 a^{2}} \sqrt[3]{2 a^{6}}$

Step 3

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

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

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Move $a^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

$2 a^{3} \sqrt[3]{2 a^{2}} \sqrt[3]{2}$

Step 6

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

$2 a^{3} \sqrt[3]{2^{2} a^{2}}$

Step 7

Raise $2$ to the power of $2$ .

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

Simplify ( cube root of 16a^5)( cube root of 2a^6)

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