Simplify ( square root of 2ab^2)( fourth root of a^3b)

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

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

Move $a$ .

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

Reorder $2$ and $b^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Rewrite the expression using the least common index of $4$ .

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Apply the product rule to $2 a$ .

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

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

Move $a^{2}$ .

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

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

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

Add $2$ and $3$ .

$b \sqrt[4]{a^{5} b \cdot 2^{2}}$

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

$b \sqrt[4]{a^{5} b \cdot 4}$

Rewrite $a^{5} b \cdot 4$ as $a^{4} (a b \cdot 4)$ .

Factor out $a^{4}$ .

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

Reorder factors in $b (a \sqrt[4]{a b \cdot 4})$ .

$b a \sqrt[4]{4 a b}$