Simplify 2xy square root of 4x^6y^3

$2 x y \sqrt{4 x^{6} y^{3}}$

Rewrite $4 x^{6} y^{3}$ as ${(2 x^{3} y)}^{2} y$ .

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

$2 x y \sqrt{2^{2} x^{6} y^{3}}$

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

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

Factor out $y^{2}$ .

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

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

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

Pull terms out from under the radical.

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

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

Move $x^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Multiply $2$ by $2$ .

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