Let's simplify the given expression step by step:
[tex]\[
\frac{\left(4 m^2 n\right)^2}{2 m^5 n}
\][/tex]
1. First, simplify the numerator [tex]\((4 m^2 n)^2\)[/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 becomes:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
2. Next, simplify the fraction by dividing the numerator and 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]
3. Simplify each part individually:
[tex]\[
\frac{16}{2} = 8
\][/tex]
[tex]\[
\frac{m^4}{m^5} = m^{4-5} = m^{-1}
\][/tex]
[tex]\[
\frac{n^2}{n} = n^{2-1} = n
\][/tex]
4. Putting it all together:
[tex]\[
8 \cdot m^{-1} \cdot n
\][/tex]
Therefore, the simplified expression is:
[tex]\[
8 m^{-1} n
\][/tex]
The correct answer is:
[tex]\[
\boxed{8 m^{-1} n}
\][/tex]
So, the correct option is C: [tex]\(8 m^{-1} n\)[/tex].
