To solve the expression [tex]\(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}\)[/tex], let's break it down step by step:
1. Simplify the inside of the expression:
[tex]\[
\frac{4 m n}{m^{-2} n^6}
\][/tex]
We can rewrite the denominator by moving [tex]\(m^{-2}\)[/tex] to the numerator where it becomes [tex]\(m^{2}\)[/tex]:
[tex]\[
= \frac{4 m n \cdot m^2}{n^6}
\][/tex]
Combine the powers of [tex]\(m\)[/tex] in the numerator:
[tex]\[
= \frac{4 m^{1 + 2} n}{n^6}
\][/tex]
Simplify the powers:
[tex]\[
= \frac{4 m^3 n}{n^6}
\][/tex]
Now, combine the powers of [tex]\(n\)[/tex] in the denominator:
[tex]\[
= 4 m^3 n^{1 - 6}
\][/tex]
Simplify the powers:
[tex]\[
= 4 m^3 n^{-5}
\][/tex]
2. Raise the entire expression to the power of -2:
[tex]\[
\left( 4 m^3 n^{-5} \right)^{-2}
\][/tex]
When we raise a product to a power, we raise each factor to that power:
[tex]\[
= 4^{-2} (m^3)^{-2} (n^{-5})^{-2}
\][/tex]
Calculate each term raised to -2:
[tex]\[
4^{-2} = \frac{1}{16}
\][/tex]
[tex]\[
(m^3)^{-2} = m^{-6} \quad \text{(since \((a^b)^c = a^{bc}\))}
\][/tex]
[tex]\[
(n^{-5})^{-2} = n^{10} \quad \text{(since \((a^b)^c = a^{bc}\))}
\][/tex]
Put these terms together:
[tex]\[
= \frac{1}{16} \cdot m^{-6} \cdot n^{10}
\][/tex]
Combine them into a single fraction:
[tex]\[
= \frac{n^{10}}{16 m^6}
\][/tex]
3. Conclusion:
The expression [tex]\(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}\)[/tex] simplifies to [tex]\(\frac{n^{10}}{16 m^6}\)[/tex].
From the given options, the correct one is:
[tex]\[
\boxed{\frac{n^{10}}{16 m^6}}
\][/tex]
