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 given expression step by step.
First, simplify the inner fraction:
[tex]\[
\frac{4 m n}{m^{-2} n^6}
\][/tex]
Recall that [tex]\(m^{-2}\)[/tex] is the same as [tex]\(\frac{1}{m^2}\)[/tex]. So the expression becomes:
[tex]\[
\frac{4 m n}{\frac{1}{m^2} n^6} = 4 m n \cdot m^2 n^{-6}
\][/tex]
Combine the exponents for like bases:
[tex]\[
= 4 m^{1+2} n^{1-6} = 4 m^3 n^{-5}
\][/tex]
Now, raise this simplified expression to the power of [tex]\(-2\)[/tex]:
[tex]\[
\left(4 m^3 n^{-5}\right)^{-2}
\][/tex]
This is the same as taking the reciprocal and raising it to the positive power of 2:
[tex]\[
= \frac{1}{\left(4 m^3 n^{-5}\right)^2}
\][/tex]
Separate the terms within the parenthesis and raise each to the power of 2:
[tex]\[
= \frac{1}{4^2 \cdot (m^3)^2 \cdot (n^{-5})^2}
\][/tex]
Calculate each exponentiated term:
[tex]\[
= \frac{1}{16 \cdot m^6 \cdot n^{-10}}
\][/tex]
Simplify the negative exponent on [tex]\(n\)[/tex]:
[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]
