The problem is to simplify the expression $((-2xy^2)^2) (\frac{x^6}{(2x)^2})^3$.

1. Problem Description

The problem is to simplify the expression $((-2xy^2)^2) (\frac{x^6}{(2x)^2})^3$.

2. Solution Steps

First, simplify the term $(-2xy^2)^2$.

$(-2xy^2)^2 = (-2)^2 x^2 (y^2)^2 = 4x^2y^4$

Next, simplify the term $(\frac{x^6}{(2x)^2})^3$.

$(\frac{x^6}{(2x)^2})^3 = (\frac{x^6}{4x^2})^3 = (\frac{1}{4}x^{6-2})^3 = (\frac{1}{4}x^4)^3 = (\frac{1}{4})^3 (x^4)^3 = \frac{1}{64}x^{12}$

Now, multiply the two simplified terms:

$(4x^2y^4) (\frac{1}{64}x^{12}) = \frac{4}{64} x^{2+12} y^4 = \frac{1}{16}x^{14}y^4$

3. Final Answer

The final answer is $\frac{1}{16}x^{14}y^4$.