Simplify x^2 square root of 8y^2

$x^{2} \sqrt{8 y^{2}}$

Rewrite $8 y^{2}$ as ${(2 y)}^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Move $2$ .

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

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

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

Pull terms out from under the radical.

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

Rewrite using the commutative property of multiplication.

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