To simplify the given expression:
[tex]\[
\frac{(4 m^2 n)^2}{2 m^5 n}
\][/tex]
Let's break it down step by step.
1. Expand the numerator:
[tex]\((4 m^2 n)^2\)[/tex] means we need to square [tex]\(4\)[/tex], [tex]\(m^2\)[/tex], and [tex]\(n\)[/tex]:
[tex]\[
(4 m^2 n)^2 = 4^2 \cdot (m^2)^2 \cdot n^2 = 16 \cdot m^4 \cdot n^2
\][/tex]
So, the expression now looks like:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
2. Simplify the fraction:
Divide each term in the numerator by the corresponding term in the denominator:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n} = \frac{16}{2} \cdot \frac{m^4}{m^5} \cdot \frac{n^2}{n}
\][/tex]
Simplify each part:
[tex]\[
\frac{16}{2} = 8,
\quad
\frac{m^4}{m^5} = m^{4 - 5} = m^{-1},
\quad
\text{and} \quad
\frac{n^2}{n} = n^{2 - 1} = n
\][/tex]
3. Combine the simplified parts:
[tex]\[
8 \cdot m^{-1} \cdot n = 8 m^{-1} n
\][/tex]
Therefore, the expression equivalent to the given expression is:
[tex]\[
8 m^{-1} n
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
