Simplify the expression $\frac{(2mn^{-2})^4}{2m^4n^3 \cdot 2m^{-4}}$.

1. Problem Description

Simplify the expression $\frac{(2mn^{-2})^4}{2m^4n^3 \cdot 2m^{-4}}$.

2. Solution Steps

First, we simplify the numerator:

$(2mn^{-2})^4 = 2^4 m^4 (n^{-2})^4 = 16m^4n^{-8}$

Next, we simplify the denominator:

$2m^4n^3 \cdot 2m^{-4} = 2 \cdot 2 \cdot m^4 \cdot m^{-4} \cdot n^3 = 4m^{4+(-4)}n^3 = 4m^0n^3 = 4 \cdot 1 \cdot n^3 = 4n^3$

Now, we divide the numerator by the denominator:

$\frac{16m^4n^{-8}}{4n^3} = \frac{16}{4} \cdot \frac{m^4}{1} \cdot \frac{n^{-8}}{n^3} = 4m^4n^{-8-3} = 4m^4n^{-11}$

We can rewrite $n^{-11}$ as $\frac{1}{n^{11}}$.

Therefore, the simplified expression is $\frac{4m^4}{n^{11}}$.

3. Final Answer

$\frac{4m^4}{n^{11}}$