Simplify 3 square root of ab*2 fourth root of ab

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

Step 1

Multiply $2$ by $3$ .

$6 \sqrt{a b} \sqrt[4]{a b}$

Step 2

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

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

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

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

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

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

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

Step 3

Combine using the product rule for radicals.

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

Step 4

Apply the product rule to $a b$ .

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

Step 5

Combine exponents.

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

Move $a^{2}$ .

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

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

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

$6 \sqrt[4]{a^{2} a^{1} b \cdot b^{2}}$

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

$6 \sqrt[4]{a^{2 + 1} b \cdot b^{2}}$

Add $2$ and $1$ .

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

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

Move $b^{2}$ .

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

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

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

$6 \sqrt[4]{a^{3} (b^{2} b^{1})}$

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

$6 \sqrt[4]{a^{3} b^{2 + 1}}$

Add $2$ and $1$ .

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