Simplify 3 square root of a^5b^7*2 square root of 5a^7b^3

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

Rewrite $a^{5} b^{7}$ as ${(a^{2} b^{3})}^{2} (a b)$ .

Factor out $a^{4}$ .

$3 \sqrt{a^{4} a b^{7}} \cdot (2 \sqrt{5 a^{7} b^{3}})$

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

$3 \sqrt{{(a^{2})}^{2} a b^{7}} \cdot (2 \sqrt{5 a^{7} b^{3}})$

Factor out $b^{6}$ .

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

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

$3 \sqrt{{(a^{2})}^{2} a ({(b^{3})}^{2} b)} \cdot (2 \sqrt{5 a^{7} b^{3}})$

Move $a$ .

$3 \sqrt{{(a^{2})}^{2} {(b^{3})}^{2} a b} \cdot (2 \sqrt{5 a^{7} b^{3}})$

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

$3 \sqrt{{(a^{2} b^{3})}^{2} a b} \cdot (2 \sqrt{5 a^{7} b^{3}})$

Add parentheses.

$3 \sqrt{{(a^{2} b^{3})}^{2} (a b)} \cdot (2 \sqrt{5 a^{7} b^{3}})$

Pull terms out from under the radical.

$3 (a^{2} b^{3} \sqrt{a b}) \cdot (2 \sqrt{5 a^{7} b^{3}})$

Rewrite $5 a^{7} b^{3}$ as ${(a^{3} b)}^{2} \cdot (5 a b)$ .

Factor out $a^{6}$ .

$3 a^{2} b^{3} \sqrt{a b} \cdot (2 \sqrt{5 (a^{6} a) b^{3}})$

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

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

Factor out $b^{2}$ .

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

Move $a$ .

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

Move $5$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Move $a^{3}$ .

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

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

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

Add $3$ and $2$ .

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

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

Move $b$ .

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

Multiply $b$ by $b^{3}$ .

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

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

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

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

Add $1$ and $3$ .

$3 a^{5} b^{4} \sqrt{a b} \cdot (2 (\sqrt{5 a b}))$

Multiply $3 a^{5} b^{4} \sqrt{a b} (2 \sqrt{5 a b})$ .

Multiply $2$ by $3$ .

$6 a^{5} b^{4} \sqrt{a b} \sqrt{5 a b}$

Combine using the product rule for radicals.

$6 a^{5} b^{4} \sqrt{5 a b (a b)}$

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

$6 a^{5} b^{4} \sqrt{5 (a^{1} a) b \cdot b}$

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

$6 a^{5} b^{4} \sqrt{5 (a^{1} a^{1}) b \cdot b}$

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

$6 a^{5} b^{4} \sqrt{5 a^{1 + 1} b \cdot b}$

Add $1$ and $1$ .

$6 a^{5} b^{4} \sqrt{5 a^{2} b \cdot b}$

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

$6 a^{5} b^{4} \sqrt{5 a^{2} (b^{1} b)}$

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

$6 a^{5} b^{4} \sqrt{5 a^{2} (b^{1} b^{1})}$

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

$6 a^{5} b^{4} \sqrt{5 a^{2} b^{1 + 1}}$

Add $1$ and $1$ .

$6 a^{5} b^{4} \sqrt{5 a^{2} b^{2}}$

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

Move $5$ .

$6 a^{5} b^{4} \sqrt{a^{2} b^{2} \cdot 5}$

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

$6 a^{5} b^{4} \sqrt{{(a b)}^{2} \cdot 5}$

Pull terms out from under the radical.

$6 a^{5} b^{4} (a b \sqrt{5})$

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

Move $a$ .

$6 (a \cdot a^{5}) b^{4} (b \sqrt{5})$

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

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

$6 (a^{1} a^{5}) b^{4} (b \sqrt{5})$

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

$6 a^{1 + 5} b^{4} (b \sqrt{5})$

Add $1$ and $5$ .

$6 a^{6} b^{4} (b \sqrt{5})$

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

Move $b$ .

$6 a^{6} (b \cdot b^{4}) \sqrt{5}$

Multiply $b$ by $b^{4}$ .

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

$6 a^{6} (b^{1} b^{4}) \sqrt{5}$

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

$6 a^{6} b^{1 + 4} \sqrt{5}$

Add $1$ and $4$ .

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

Simplify 3 square root of a^5b^7*2 square root of 5a^7b^3

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