Select the correct answer.

Which expression is equivalent to the given expression?

[tex]\[ \frac{\left(4 m^2 n\right)^2}{2 m^2 n} \][/tex]

A. \(8 m^9 n^3\)
B. \(4 m^9 n^3\)
C. \(8 m^{-1} n\)
D. [tex]\(4 m^{-1} n\)[/tex]



Answer :

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.