Simplify 2ab^2 square root of (200a^2b^5c^11)/(c^11)

$2 a b^{2} \sqrt{\frac{200 a^{2} b^{5} c^{11}}{c^{11}}}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{200 a^{2} b^{5} c^{11}}{c^{11}}$ by cancelling the common factors.

Cancel the common factor.

$2 a b^{2} \sqrt{\frac{200 a^{2} b^{5} c^{11}}{c^{11}}}$

Rewrite the expression.

$2 a b^{2} \sqrt{\frac{200 a^{2} b^{5}}{1}}$

Divide $200 a^{2} b^{5}$ by $1$ .

$2 a b^{2} \sqrt{200 a^{2} b^{5}}$

Rewrite $200 a^{2} b^{5}$ as ${(10 a b^{2})}^{2} \cdot (2 b)$ .

Factor $100$ out of $200$ .

$2 a b^{2} \sqrt{100 (2) a^{2} b^{5}}$

Rewrite $100$ as $10^{2}$ .

$2 a b^{2} \sqrt{10^{2} \cdot 2 a^{2} b^{5}}$

Factor out $b^{4}$ .

$2 a b^{2} \sqrt{10^{2} \cdot 2 a^{2} (b^{4} b)}$

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

$2 a b^{2} \sqrt{10^{2} \cdot 2 a^{2} ({(b^{2})}^{2} b)}$

Move $2$ .

$2 a b^{2} \sqrt{10^{2} a^{2} {(b^{2})}^{2} \cdot 2 b}$

Rewrite $10^{2} a^{2} {(b^{2})}^{2}$ as ${(10 a b^{2})}^{2}$ .

$2 a b^{2} \sqrt{{(10 a b^{2})}^{2} \cdot 2 b}$

Add parentheses.

$2 a b^{2} \sqrt{{(10 a b^{2})}^{2} \cdot (2 b)}$

Pull terms out from under the radical.

$2 a b^{2} (10 a b^{2} \sqrt{2 b})$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$2 (a \cdot a) b^{2} (10 b^{2} \sqrt{2 b})$

Multiply $a$ by $a$ .

$2 a^{2} b^{2} (10 b^{2} \sqrt{2 b})$

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

Move $b^{2}$ .

$2 a^{2} (b^{2} b^{2}) (10 \sqrt{2 b})$

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

$2 a^{2} b^{2 + 2} (10 \sqrt{2 b})$

Add $2$ and $2$ .

$2 a^{2} b^{4} (10 \sqrt{2 b})$

Multiply $10$ by $2$ .

$20 a^{2} b^{4} \sqrt{2 b}$