Select the correct answer.

Which expression is equivalent to the given expression?
[tex]\[
\frac{\left(4 m^2 n\right)^2}{2 m^5 n}
\][/tex]

A. [tex]\(8 m^9 n^3\)[/tex]

B. [tex]\(4 m^9 n^3\)[/tex]

C. [tex]\(8 m^{-1} n\)[/tex]

D. [tex]\(4 m^{-1} n\)[/tex]



Answer :

Let's simplify the given expression step by step:
[tex]\[ \frac{\left(4 m^2 n\right)^2}{2 m^5 n} \][/tex]

1. First, simplify the numerator [tex]\((4 m^2 n)^2\)[/tex]:
[tex]\[ (4 m^2 n)^2 = 4^2 \cdot (m^2)^2 \cdot n^2 = 16 \cdot m^4 \cdot n^2 \][/tex]

So the expression becomes:
[tex]\[ \frac{16 m^4 n^2}{2 m^5 n} \][/tex]

2. Next, simplify the fraction by dividing the numerator and the denominator:
[tex]\[ \frac{16 m^4 n^2}{2 m^5 n} = \frac{16}{2} \cdot \frac{m^4}{m^5} \cdot \frac{n^2}{n} \][/tex]

3. Simplify each part individually:
[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]

4. Putting it all together:
[tex]\[ 8 \cdot m^{-1} \cdot n \][/tex]

Therefore, the simplified expression is:
[tex]\[ 8 m^{-1} n \][/tex]

The correct answer is:
[tex]\[ \boxed{8 m^{-1} n} \][/tex]
So, the correct option is C: [tex]\(8 m^{-1} n\)[/tex].