To find an equivalent expression for [tex]\(\frac{(4 m^2 n)^2}{2 m^5 n}\)[/tex], let's simplify the given expression step-by-step.
1. First, let's expand the numerator [tex]\((4 m^2 n)^2\)[/tex]:
[tex]\[
(4 m^2 n)^2 = 4^2 \cdot (m^2)^2 \cdot n^2
\][/tex]
Simplifying further:
[tex]\[
(4^2) = 16, \quad (m^2)^2 = m^{2 \cdot 2} = m^4, \quad n^2
\][/tex]
So the expanded numerator becomes:
[tex]\[
16 m^4 n^2
\][/tex]
2. Now, let's write the entire expression with the expanded numerator:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
3. Next, let's simplify the fraction by dividing the coefficients:
[tex]\[
\frac{16}{2} = 8
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{8 m^4 n^2}{m^5 n}
\][/tex]
4. Now, simplify the powers of [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
- For the powers of [tex]\(m\)[/tex]:
[tex]\[
m^4 \div m^5 = m^{4-5} = m^{-1}
\][/tex]
- For the powers of [tex]\(n\)[/tex]:
[tex]\[
n^2 \div n = n^{2-1} = n^{1} = n
\][/tex]
So, the expression simplifies to:
[tex]\[
8 m^{-1} n
\][/tex]
5. Finally, let's write the simplified expression clearly:
[tex]\[
8 \frac{n}{m}
\][/tex]
From this detailed solution, it's clear that the expression [tex]\(\frac{(4 m^2 n)^2}{2 m^5 n}\)[/tex] simplifies to [tex]\(8 m^{-1} n\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{8 m^{-1} n}
\][/tex]
This matches option B.
