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

1. Problem Description

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

2. Solution Steps

First, we simplify the numerator:

$(-2a^2b^4)^2 = (-2)^2(a^2)^2(b^4)^2$

$= 4a^{2*2}b^{4*2}$

$= 4a^4b^8$

So, the expression becomes $\frac{4a^4b^8}{12a^3b^2}$.

Now, simplify the fraction:

$\frac{4}{12} = \frac{1}{3}$

$\frac{a^4}{a^3} = a^{4-3} = a^1 = a$

$\frac{b^8}{b^2} = b^{8-2} = b^6$

Therefore, $\frac{4a^4b^8}{12a^3b^2} = \frac{1}{3}ab^6$.

3. Final Answer

$\frac{1}{3}ab^6$