Find the Product square root of 2a^3b^3* cube root of 4ab^2

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

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

Factor out $a^{2}$ .

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

Factor out $b^{2}$ .

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

Move $a$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 a b^{2}}$ as ${(4 a b^{2})}^{\frac{1}{3}}$ .

$a b ({(4 a b^{2})}^{\frac{1}{3}} \sqrt{2 a b})$

Rewrite ${(4 a b^{2})}^{\frac{1}{3}}$ as ${(4 a b^{2})}^{\frac{2}{6}}$ .

$a b ({(4 a b^{2})}^{\frac{2}{6}} \sqrt{2 a b})$

Rewrite ${(4 a b^{2})}^{\frac{2}{6}}$ as $\sqrt[6]{{(4 a b^{2})}^{2}}$ .

$a b (\sqrt[6]{{(4 a b^{2})}^{2}} \sqrt{2 a b})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 a b}$ as ${(2 a b)}^{\frac{1}{2}}$ .

$a b (\sqrt[6]{{(4 a b^{2})}^{2}} {(2 a b)}^{\frac{1}{2}})$

Rewrite ${(2 a b)}^{\frac{1}{2}}$ as ${(2 a b)}^{\frac{3}{6}}$ .

$a b (\sqrt[6]{{(4 a b^{2})}^{2}} {(2 a b)}^{\frac{3}{6}})$

Rewrite ${(2 a b)}^{\frac{3}{6}}$ as $\sqrt[6]{{(2 a b)}^{3}}$ .

$a b (\sqrt[6]{{(4 a b^{2})}^{2}} \sqrt[6]{{(2 a b)}^{3}})$

Combine using the product rule for radicals.

$a b \sqrt[6]{{(4 a b^{2})}^{2} {(2 a b)}^{3}}$

Apply the product rule to $4 a b^{2}$ .

$a b \sqrt[6]{{(4 a)}^{2} {(b^{2})}^{2} {(2 a b)}^{3}}$

Apply the product rule to $2 a b$ .

$a b \sqrt[6]{{(4 a)}^{2} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}$

Apply the product rule to $4 a$ .

$a b \sqrt[6]{4^{2} a^{2} {(b^{2})}^{2} {(2 a)}^{3} b^{3}}$

Raise $4$ to the power of $2$ .

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

Multiply the exponents in ${(b^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $2$ .

$a b \sqrt[6]{16 a^{2} b^{4} {(2 a)}^{3} b^{3}}$

Apply the product rule to $2 a$ .

$a b \sqrt[6]{16 a^{2} b^{4} (2^{3} a^{3}) b^{3}}$

Raise $2$ to the power of $3$ .

$a b \sqrt[6]{16 a^{2} b^{4} \cdot 8 a^{3} b^{3}}$

Combine exponents.

Multiply $8$ by $16$ .

$a b \sqrt[6]{128 a^{2} b^{4} a^{3} b^{3}}$

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$a b \sqrt[6]{128 (a^{3} a^{2}) b^{4} b^{3}}$

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

$a b \sqrt[6]{128 a^{3 + 2} b^{4} b^{3}}$

Add $3$ and $2$ .

$a b \sqrt[6]{128 a^{5} b^{4} b^{3}}$

Multiply $b^{4}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

$a b \sqrt[6]{128 a^{5} (b^{3} b^{4})}$

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

$a b \sqrt[6]{128 a^{5} b^{3 + 4}}$

Add $3$ and $4$ .

$a b \sqrt[6]{128 a^{5} b^{7}}$

Rewrite $128 a^{5} b^{7}$ as ${(2 b)}^{6} \cdot (2 a^{5} b)$ .

Factor $64$ out of $128$ .

$a b \sqrt[6]{64 (2) a^{5} b^{7}}$

Rewrite $64$ as $2^{6}$ .

$a b \sqrt[6]{2^{6} \cdot 2 a^{5} b^{7}}$

Factor out $b^{6}$ .

$a b \sqrt[6]{2^{6} \cdot 2 a^{5} (b^{6} b)}$

Move $a^{5}$ .

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

Move $2$ .

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

Rewrite $2^{6} b^{6}$ as ${(2 b)}^{6}$ .

$a b \sqrt[6]{{(2 b)}^{6} \cdot 2 a^{5} b}$

Add parentheses.

$a b \sqrt[6]{{(2 b)}^{6} \cdot 2 (a^{5} b)}$

Add parentheses.

$a b \sqrt[6]{{(2 b)}^{6} \cdot (2 a^{5} b)}$

Pull terms out from under the radical.

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

Rewrite using the commutative property of multiplication.

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

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

Move $b$ .

$2 a (b \cdot b) \sqrt[6]{2 a^{5} b}$

Multiply $b$ by $b$ .

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

Find the Product square root of 2a^3b^3* cube root of 4ab^2

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