To simplify the given expression:
[tex]\[
\frac{\left(4 m^2 n\right)^2}{2 m^2 n}
\][/tex]
we'll follow a step-by-step approach.
1. Simplify the numerator:
The numerator is \((4 m^2 n)^2\).
Raise each part of the expression inside the parentheses to the power of 2:
[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 numerator becomes \(16 m^4 n^2\).
2. Simplify the denominator:
The denominator is \(2 m^2 n\).
It remains \(2 m^2 n\).
3. Divide the numerator by the denominator:
[tex]\[
\frac{16 m^4 n^2}{2 m^2 n}
\][/tex]
Separate the constants, \(m\) terms, and \(n\) terms:
[tex]\[
\frac{16}{2} \cdot \frac{m^4}{m^2} \cdot \frac{n^2}{n}
\][/tex]
Simplify each fraction:
- \(\frac{16}{2} = 8\)
- \(\frac{m^4}{m^2} = m^{4-2} = m^2\)
- \(\frac{n^2}{n} = n^{2-1} = n\)
Combine the simplified parts:
[tex]\[
8 \cdot m^2 \cdot n = 8 m^2 n
\][/tex]
So, the expression equivalent to the given expression is:
[tex]\[
8 m^2 n
\][/tex]
Since none of the given answers directly match \(8 m^2 n\), it seems there is either an error in the transcription of the question or answers or an inconsistency in the provided choices. Based on the detailed simplification, the correct answer is:
[tex]\[
8 m^2 n
\][/tex]
But, if we must choose from the given options, none of the provided options (A, B, C, or D) are correct.
