To determine which expression is equivalent to [tex]\(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}\)[/tex], we need to simplify the expression step by step.
1. Simplifying the inner fraction:
[tex]\[
\frac{4 m n}{m^{-2} n^6}
\][/tex]
First, handle the negative exponent in the denominator. Recall that [tex]\(m^{-2} = \frac{1}{m^2}\)[/tex]:
[tex]\[
\frac{4 m n}{\frac{1}{m^2} n^6} = 4 m n \cdot \frac{m^2}{n^6} = 4 m^{1+2} \cdot \frac{n}{n^6} = 4 m^3 \cdot \frac{n}{n^6} = 4 m^3 \cdot n^{1-6} = 4 m^3 \cdot n^{-5}
\][/tex]
So we have:
[tex]\[
4 m^3 n^{-5}
\][/tex]
2. Applying the negative exponent to the simplified expression:
[tex]\[
\left(4 m^3 n^{-5}\right)^{-2}
\][/tex]
Recall the rule [tex]\((a \cdot b)^{-c} = a^{-c} \cdot b^{-c}\)[/tex]:
[tex]\[
\left(4 \cdot m^3 \cdot n^{-5}\right)^{-2} = 4^{-2} \cdot (m^3)^{-2} \cdot (n^{-5})^{-2}
\][/tex]
Simplify each part:
[tex]\[
4^{-2} = \frac{1}{4^2} = \frac{1}{16}
\][/tex]
[tex]\[
(m^3)^{-2} = m^{3 \cdot -2} = m^{-6}
\][/tex]
[tex]\[
(n^{-5})^{-2} = n^{-5 \cdot -2} = n^{10}
\][/tex]
Combine these results:
[tex]\[
\frac{1}{16} \cdot m^{-6} \cdot n^{10} = \frac{n^{10}}{16 m^6}
\][/tex]
Therefore, the expression equivalent to [tex]\(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}\)[/tex] is:
[tex]\[
\boxed{\frac{n^{10}}{16 m^6}}
\][/tex]
