To simplify the given expression, [tex]\(\frac{(4m^2n)^2}{2m^3n}\)[/tex], let's break it down step-by-step:
1. Simplify the numerator:
[tex]\[
(4m^2n)^2
\][/tex]
When raising a product to a power, raise each factor to the power:
[tex]\[
(4)^2 \cdot (m^2)^2 \cdot (n)^2
\][/tex]
Calculate each term:
[tex]\[
16m^4n^2
\][/tex]
2. Simplify the denominator:
[tex]\[
2m^3n
\][/tex]
3. Combine the fraction:
[tex]\[
\frac{16m^4n^2}{2m^3n}
\][/tex]
4. Simplify the coefficients:
[tex]\[
\frac{16}{2} = 8
\][/tex]
5. Simplify the [tex]\(m\)[/tex]-terms:
[tex]\[
\frac{m^4}{m^3} = m^{4-3} = m
\][/tex]
6. Simplify the [tex]\(n\)[/tex]-terms:
[tex]\[
\frac{n^2}{n} = n^{2-1} = n
\][/tex]
Putting it all together:
[tex]\[
8mn
\][/tex]
Thus, the expression [tex]\(\frac{(4m^2n)^2}{2m^3n}\)[/tex] simplifies to [tex]\(8mn\)[/tex], which corresponds to [tex]\(8m^1n^1\)[/tex].
Given the options, the correct answer is:
A. [tex]\(8m^{-1}n\)[/tex]
B. [tex]\(4m^{-1}n\)[/tex]
C. [tex]\(8m^9n^3\)[/tex]
D. [tex]\(4m^9n^3\)[/tex]
So the correct answer is:
None of the above are correct as `8 m n is not present in the given options.`
