Simplify square root of (2x)^2 square root of 2y

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

Apply the product rule to $2 x$ .

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

Raise $2$ to the power of $2$ .

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

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

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

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

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

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

Pull terms out from under the radical.

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

Rewrite $\sqrt{\sqrt{2 y}}$ as $\sqrt[4]{2 y}$ .

$2 x \sqrt[4]{2 y}$