Simplify 2 square root of 32a^3b^5* square root of 8a^7b^2

$2 \sqrt{32 a^{3} b^{5}} \cdot \sqrt{8 a^{7} b^{2}}$

Step 1

Rewrite $32 a^{3} b^{5}$ as ${(4 a b^{2})}^{2} \cdot (2 a b)$ .

Factor $16$ out of $32$ .

$2 \sqrt{16 (2) a^{3} b^{5}} \cdot \sqrt{8 a^{7} b^{2}}$

Rewrite $16$ as $4^{2}$ .

$2 \sqrt{4^{2} \cdot 2 a^{3} b^{5}} \cdot \sqrt{8 a^{7} b^{2}}$

Factor out $a^{2}$ .

$2 \sqrt{4^{2} \cdot 2 (a^{2} a) b^{5}} \cdot \sqrt{8 a^{7} b^{2}}$

Factor out $b^{4}$ .

$2 \sqrt{4^{2} \cdot 2 (a^{2} a) (b^{4} b)} \cdot \sqrt{8 a^{7} b^{2}}$

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

$2 \sqrt{4^{2} \cdot 2 (a^{2} a) ({(b^{2})}^{2} b)} \cdot \sqrt{8 a^{7} b^{2}}$

Move $a$ .

$2 \sqrt{4^{2} \cdot 2 (a^{2}) {(b^{2})}^{2} a b} \cdot \sqrt{8 a^{7} b^{2}}$

Move $2$ .

$2 \sqrt{(4^{2} (a^{2})) {(b^{2})}^{2} \cdot 2 a b} \cdot \sqrt{8 a^{7} b^{2}}$

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

$2 \sqrt{{(4 a b^{2})}^{2} \cdot 2 a b} \cdot \sqrt{8 a^{7} b^{2}}$

Add parentheses.

$2 \sqrt{{(4 a b^{2})}^{2} \cdot 2 (a b)} \cdot \sqrt{8 a^{7} b^{2}}$

Add parentheses.

$2 \sqrt{{(4 a b^{2})}^{2} \cdot (2 a b)} \cdot \sqrt{8 a^{7} b^{2}}$

Step 2

Pull terms out from under the radical.

$2 (4 a b^{2} \sqrt{2 a b}) \cdot \sqrt{8 a^{7} b^{2}}$

Step 3

Multiply $4$ by $2$ .

$8 (a b^{2} \sqrt{2 a b}) \cdot \sqrt{8 a^{7} b^{2}}$

Step 4

Rewrite $8 a^{7} b^{2}$ as ${(2 a^{3} b)}^{2} \cdot (2 a)$ .

Factor $4$ out of $8$ .

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{4 (2) a^{7} b^{2}}$

Rewrite $4$ as $2^{2}$ .

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{2^{2} \cdot 2 a^{7} b^{2}}$

Factor out $a^{6}$ .

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{2^{2} \cdot 2 (a^{6} a) b^{2}}$

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

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{2^{2} \cdot 2 ({(a^{3})}^{2} a) b^{2}}$

Move $a$ .

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{2^{2} \cdot 2 {(a^{3})}^{2} b^{2} a}$

Move $2$ .

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{2^{2} {(a^{3})}^{2} b^{2} \cdot 2 a}$

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

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{{(2 a^{3} b)}^{2} \cdot 2 a}$

Add parentheses.

$8 a b^{2} \sqrt{2 a b} \cdot \sqrt{{(2 a^{3} b)}^{2} \cdot (2 a)}$

Step 5

Pull terms out from under the radical.

$8 a b^{2} \sqrt{2 a b} \cdot (2 a^{3} b \sqrt{2 a})$

Step 6

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

Move $a^{3}$ .

$8 (a^{3} a) b^{2} \sqrt{2 a b} \cdot (2 b \sqrt{2 a})$

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

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

$8 (a^{3} a^{1}) b^{2} \sqrt{2 a b} \cdot (2 b \sqrt{2 a})$

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

$8 a^{3 + 1} b^{2} \sqrt{2 a b} \cdot (2 b \sqrt{2 a})$

Add $3$ and $1$ .

$8 a^{4} b^{2} \sqrt{2 a b} \cdot (2 b \sqrt{2 a})$

Step 7

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

Move $b$ .

$8 a^{4} (b \cdot b^{2}) \sqrt{2 a b} \cdot (2 \sqrt{2 a})$

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

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

$8 a^{4} (b^{1} b^{2}) \sqrt{2 a b} \cdot (2 \sqrt{2 a})$

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

$8 a^{4} b^{1 + 2} \sqrt{2 a b} \cdot (2 \sqrt{2 a})$

Add $1$ and $2$ .

$8 a^{4} b^{3} \sqrt{2 a b} \cdot (2 \sqrt{2 a})$

Step 8

Multiply $8 a^{4} b^{3} \sqrt{2 a b} (2 \sqrt{2 a})$ .

Multiply $2$ by $8$ .

$16 a^{4} b^{3} \sqrt{2 a b} \sqrt{2 a}$

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

$16 a^{4} b^{3} \sqrt{4 a (a b)}$

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

$16 a^{4} b^{3} \sqrt{4 (a^{1} a) b}$

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

$16 a^{4} b^{3} \sqrt{4 (a^{1} a^{1}) b}$

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

$16 a^{4} b^{3} \sqrt{4 a^{1 + 1} b}$

Add $1$ and $1$ .

$16 a^{4} b^{3} \sqrt{4 a^{2} b}$

Step 9

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

Rewrite $4$ as $2^{2}$ .

$16 a^{4} b^{3} \sqrt{2^{2} a^{2} b}$

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

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

Step 10

Pull terms out from under the radical.

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

Step 11

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

Move $a$ .

$16 (a \cdot a^{4}) b^{3} (2 \sqrt{b})$

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

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

$16 (a^{1} a^{4}) b^{3} (2 \sqrt{b})$

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

$16 a^{1 + 4} b^{3} (2 \sqrt{b})$

Add $1$ and $4$ .

$16 a^{5} b^{3} (2 \sqrt{b})$

Step 12

Multiply $2$ by $16$ .

$32 a^{5} b^{3} \sqrt{b}$

Simplify 2 square root of 32a^3b^5* square root of 8a^7b^2

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