Let's solve the given expression step by step to find the equivalent expression. The expression provided is:
[tex]\[
\frac{(4 m^2 n)^2}{2 m^5 n}
\][/tex]
First, we need to simplify the numerator [tex]\((4 m^2 n)^2\)[/tex]:
[tex]\[
(4 m^2 n)^2 = (4)^2 \cdot (m^2)^2 \cdot (n)^2 = 16 m^4 n^2
\][/tex]
So, the expression now looks like:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
Next, we simplify this fraction by dividing both the numerator and the denominator by common factors.
Start by simplifying the constants:
[tex]\[
\frac{16}{2} = 8
\][/tex]
Now we have:
[tex]\[
\frac{8 m^4 n^2}{m^5 n}
\][/tex]
To further simplify this, we divide each variable (and their exponents) in the numerator by the corresponding variable (and their exponents) in the denominator.
For [tex]\(m\)[/tex]:
[tex]\[
\frac{m^4}{m^5} = m^{4-5} = m^{-1}
\][/tex]
For [tex]\(n\)[/tex]:
[tex]\[
\frac{n^2}{n} = n^{2-1} = n
\][/tex]
Putting it all together, we have:
[tex]\[
8 m^{-1} n
\][/tex]
Hence, the correct expression equivalent to the given expression is:
[tex]\[
8 m^{-1} n
\][/tex]
The correct choice is:
A. [tex]\(8 m^{-1} n\)[/tex]
