Simplify square root of 3a^2b^5 square root of 6ab^7

$\sqrt{3 a^{2} b^{5}} \sqrt{6 a b^{7}}$

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

Factor out $b^{4}$ .

$\sqrt{3 a^{2} (b^{4} b)} \sqrt{6 a b^{7}}$

Rewrite $b^{4}$ as ${(b^{2})}^{2}$ .

$\sqrt{3 a^{2} ({(b^{2})}^{2} b)} \sqrt{6 a b^{7}}$

Move $3$ .

$\sqrt{a^{2} {(b^{2})}^{2} \cdot 3 b} \sqrt{6 a b^{7}}$

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

$\sqrt{{(a b^{2})}^{2} \cdot 3 b} \sqrt{6 a b^{7}}$

Add parentheses.

$\sqrt{{(a b^{2})}^{2} \cdot (3 b)} \sqrt{6 a b^{7}}$

Pull terms out from under the radical.

$a b^{2} \sqrt{3 b} \sqrt{6 a b^{7}}$

Rewrite $6 a b^{7}$ as ${(b^{3})}^{2} \cdot (6 a b)$ .

Factor out $b^{6}$ .

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

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

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

Move $a$ .

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

Reorder $6$ and ${(b^{3})}^{2}$ .

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Move $b^{3}$ .

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

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

$a b^{3 + 2} \sqrt{3 b} \sqrt{6 a b}$

Add $3$ and $2$ .

$a b^{5} \sqrt{3 b} \sqrt{6 a b}$

Multiply $a b^{5} \sqrt{3 b} \sqrt{6 a b}$ .

Combine using the product rule for radicals.

$a b^{5} \sqrt{6 a b (3 b)}$

Multiply $3$ by $6$ .

$a b^{5} \sqrt{18 a b \cdot b}$

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

$a b^{5} \sqrt{18 a (b^{1} b)}$

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

$a b^{5} \sqrt{18 a (b^{1} b^{1})}$

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

$a b^{5} \sqrt{18 a b^{1 + 1}}$

Add $1$ and $1$ .

$a b^{5} \sqrt{18 a b^{2}}$

Rewrite $18 a b^{2}$ as ${(3 b)}^{2} \cdot (2 a)$ .

Factor $9$ out of $18$ .

$a b^{5} \sqrt{9 (2) a b^{2}}$

Rewrite $9$ as $3^{2}$ .

$a b^{5} \sqrt{3^{2} \cdot 2 a b^{2}}$

Move $a$ .

$a b^{5} \sqrt{3^{2} \cdot 2 b^{2} a}$

Move $2$ .

$a b^{5} \sqrt{3^{2} b^{2} \cdot 2 a}$

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

$a b^{5} \sqrt{{(3 b)}^{2} \cdot 2 a}$

Add parentheses.

$a b^{5} \sqrt{{(3 b)}^{2} \cdot (2 a)}$

Pull terms out from under the radical.

$a b^{5} (3 b \sqrt{2 a})$

Rewrite using the commutative property of multiplication.

$3 a b^{5} (b \sqrt{2 a})$

Multiply $b^{5}$ by $b$ by adding the exponents.

Move $b$ .

$3 a (b \cdot b^{5}) \sqrt{2 a}$

Multiply $b$ by $b^{5}$ .

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

$3 a (b^{1} b^{5}) \sqrt{2 a}$

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

$3 a b^{1 + 5} \sqrt{2 a}$

Add $1$ and $5$ .

$3 a b^{6} \sqrt{2 a}$

Simplify square root of 3a^2b^5 square root of 6ab^7

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