To simplify the given expression \(\frac{\left(4 m^2 n\right)^2}{2 m^5 n}\), let's go through the steps one by one.
1. Simplify the numerator \(\left(4 m^2 n\right)^2\)
- First, distribute the exponent 2 to each factor inside the parentheses:
[tex]\[
\left(4 m^2 n\right)^2 = 4^2 \cdot (m^2)^2 \cdot n^2
\][/tex]
- Then calculate each power:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
(m^2)^2 = m^{2 \cdot 2} = m^4
\][/tex]
[tex]\[
n^2 = n^2
\][/tex]
- So the numerator becomes:
[tex]\[
16 m^4 n^2
\][/tex]
2. Rewrite the expression with the simplified numerator
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
3. Simplify the fraction
- Divide the coefficients (numbers) \(16\) and \(2\):
[tex]\[
\frac{16}{2} = 8
\][/tex]
- Subtract the exponents of \(m\):
[tex]\[
m^4 / m^5 = m^{4-5} = m^{-1}
\][/tex]
- Subtract the exponents of \(n\):
[tex]\[
n^2 / n = n^{2-1} = n^1 = n
\][/tex]
4. Combine the simplified terms
- So the expression becomes:
[tex]\[
8 \cdot m^{-1} \cdot n = \frac{8n}{m}
\][/tex]
Therefore, the expression equivalent to \(\frac{\left(4 m^2 n\right)^2}{2 m^5 n}\) is \(\frac{8 n}{m}\).
Given the multiple-choice options, the correct answer is:
C. [tex]\(8 m^{-1} n\)[/tex]
