Let's start by simplifying the given expression step-by-step:
Given expression:
[tex]\[
\frac{(4 m^2 n)^2}{2 m^5 n}
\][/tex]
1. First, simplify the numerator [tex]\((4 m^2 n)^2\)[/tex]:
[tex]\[
(4 m^2 n)^2 = (4)^2 \cdot (m^2)^2 \cdot (n)^2
\][/tex]
Calculating each part separately:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
(m^2)^2 = m^{2 \cdot 2} = m^4
\][/tex]
[tex]\[
n^2 = n^2
\][/tex]
So,
[tex]\[
(4 m^2 n)^2 = 16 m^4 n^2
\][/tex]
2. Next, substitute the simplified numerator back into the original expression:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
3. Simplify the denominator:
[tex]\[
2 m^5 n
\][/tex]
4. Now, divide each term in the numerator by the corresponding term in the denominator:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
Separate the fraction:
[tex]\[
\frac{16}{2} \cdot \frac{m^4}{m^5} \cdot \frac{n^2}{n}
\][/tex]
Calculate each part:
[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]
So,
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n} = 8 m^{-1} n
\][/tex]
Therefore, the expression equivalent to the given expression is:
B. [tex]\(8 m^{-1} n\)[/tex]
