To simplify √((2x)^2)√2y, we must multiply the indices of both roots and the coefficients that within both roots. As we can see bellow, we get a root which has index 4:
=((2x)^(2y))^1/4
Finally, the factor (2x)^2 has an exponent which is divisible by the index of the root, so it can be simplified, as it's shown in the following step:
=
(2x)(2y)^1/4