Simplify the expression $(\frac{-2a^2b^{\frac{3}{4}}}{12a^3b^{\frac{1}{2}}})^2$.

1. Problem Description

Simplify the expression

$(\frac{-2a^2b^{\frac{3}{4}}}{12a^3b^{\frac{1}{2}}})^2$.

2. Solution Steps

First, simplify the expression inside the parenthesis.

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

$= (\frac{-b^{\frac{1}{4}}}{6a})^2$

Now, apply the power of 2 to the simplified fraction:

$(\frac{-b^{\frac{1}{4}}}{6a})^2 = \frac{(-b^{\frac{1}{4}})^2}{(6a)^2} = \frac{b^{\frac{2}{4}}}{36a^2} = \frac{b^{\frac{1}{2}}}{36a^2}$

3. Final Answer

$\frac{b^{\frac{1}{2}}}{36a^2}$