Simplify 4xy^2 square root of 25x^4y^7

$4 x y^{2} \sqrt{25 x^{4} y^{7}}$

Rewrite $25 x^{4} y^{7}$ as ${(5 x^{2} y^{3})}^{2} y$ .

Rewrite $25$ as $5^{2}$ .

$4 x y^{2} \sqrt{5^{2} x^{4} y^{7}}$

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

$4 x y^{2} \sqrt{5^{2} {(x^{2})}^{2} y^{7}}$

Factor out $y^{6}$ .

$4 x y^{2} \sqrt{5^{2} {(x^{2})}^{2} (y^{6} y)}$

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

$4 x y^{2} \sqrt{5^{2} {(x^{2})}^{2} ({(y^{3})}^{2} y)}$

Rewrite $5^{2} {(x^{2})}^{2} {(y^{3})}^{2}$ as ${(5 x^{2} y^{3})}^{2}$ .

$4 x y^{2} \sqrt{{(5 x^{2} y^{3})}^{2} y}$

Pull terms out from under the radical.

$4 x y^{2} (5 x^{2} y^{3} \sqrt{y})$

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

Move $x^{2}$ .

$4 (x^{2} x) y^{2} (5 y^{3} \sqrt{y})$

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

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

$4 (x^{2} x^{1}) y^{2} (5 y^{3} \sqrt{y})$

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

$4 x^{2 + 1} y^{2} (5 y^{3} \sqrt{y})$

Add $2$ and $1$ .

$4 x^{3} y^{2} (5 y^{3} \sqrt{y})$

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

Move $y^{3}$ .

$4 x^{3} (y^{3} y^{2}) (5 \sqrt{y})$

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

$4 x^{3} y^{3 + 2} (5 \sqrt{y})$

Add $3$ and $2$ .

$4 x^{3} y^{5} (5 \sqrt{y})$

Multiply $5$ by $4$ .

$20 x^{3} y^{5} \sqrt{y}$