Simplify cube root of 4a^2b^4* cube root of 16a^3b^2

$\sqrt[3]{4 a^{2} b^{4}} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Step 1

Rewrite $4 a^{2} b^{4}$ as $b^{3} \cdot (4 a^{2} b)$ .

Factor out $b^{3}$ .

$\sqrt[3]{4 a^{2} (b^{3} b)} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Move $a^{2}$ .

$\sqrt[3]{4 b^{3} a^{2} b} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Reorder $4$ and $b^{3}$ .

$\sqrt[3]{b^{3} \cdot 4 a^{2} b} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Add parentheses.

$\sqrt[3]{b^{3} \cdot 4 (a^{2} b)} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Add parentheses.

$\sqrt[3]{b^{3} \cdot (4 a^{2} b)} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Step 2

Pull terms out from under the radical.

$b \sqrt[3]{4 a^{2} b} \cdot \sqrt[3]{16 a^{3} b^{2}}$

Step 3

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

Factor $8$ out of $16$ .

$b \sqrt[3]{4 a^{2} b} \cdot \sqrt[3]{8 (2) a^{3} b^{2}}$

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

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $2 b \sqrt[3]{4 a^{2} b} (a \sqrt[3]{2 b^{2}})$ .

Combine using the product rule for radicals.

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

Multiply $2$ by $4$ .

$2 b (a \sqrt[3]{8 a^{2} b \cdot b^{2}})$

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

Move $b^{2}$ .

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

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

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

$2 b (a \sqrt[3]{8 a^{2} (b^{2} b^{1})})$

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

$2 b (a \sqrt[3]{8 a^{2} b^{2 + 1}})$

Add $2$ and $1$ .

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

Step 7

Rewrite $8 a^{2} b^{3}$ as ${(2 b)}^{3} a^{2}$ .

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

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

Move $a^{2}$ .

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

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

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

Multiply $b$ by $b$ .

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

Step 10

Rewrite using the commutative property of multiplication.

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

Step 11

Multiply $2$ by $2$ .

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

Simplify cube root of 4a^2b^4* cube root of 16a^3b^2

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