Simplify the expression $\frac{u^2 - 4v^2}{(6u + 12v)^2}$.

1. Problem Description

Simplify the expression $\frac{u^2 - 4v^2}{(6u + 12v)^2}$.

2. Solution Steps

First, we can factor the numerator using the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$.

In this case, $a = u$ and $b = 2v$, so the numerator becomes $u^2 - 4v^2 = (u - 2v)(u + 2v)$.

Next, we can factor out a 6 from the expression in the denominator:

$6u + 12v = 6(u + 2v)$.

Then, square the expression:

$(6u + 12v)^2 = [6(u + 2v)]^2 = 6^2(u + 2v)^2 = 36(u + 2v)^2$.

Now, rewrite the original expression:

$\frac{u^2 - 4v^2}{(6u + 12v)^2} = \frac{(u - 2v)(u + 2v)}{36(u + 2v)^2}$

We can simplify the expression by canceling out the $(u + 2v)$ term in the numerator and denominator:

$\frac{(u - 2v)(u + 2v)}{36(u + 2v)(u + 2v)} = \frac{u - 2v}{36(u + 2v)}$

3. Final Answer

$\frac{u - 2v}{36(u + 2v)}$