Simplify the expression $((-2xy^2)^2 (\frac{x^6}{(2x)^2})^3)$.

1. Problem Description

Simplify the expression $((-2xy^2)^2 (\frac{x^6}{(2x)^2})^3)$.

2. Solution Steps

First, simplify the expression $(-2xy^2)^2$:

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

Next, simplify $(\frac{x^6}{(2x)^2})^3$:

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

Now multiply the two simplified expressions together:

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

3. Final Answer

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