To determine which expression is equivalent to [tex]\(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}\)[/tex], let's go through the problem step by step.
Given expression:
[tex]\[
\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}
\][/tex]
First, let's simplify the fraction inside the parentheses:
[tex]\[
\frac{4 m n}{m^{-2} n^6}
\][/tex]
Recall that [tex]\( m^{-2} = \frac{1}{m^2} \)[/tex], so we can rewrite the denominator:
[tex]\[
\frac{4 m n}{\frac{1}{m^2} n^6} = 4 m n \cdot \frac{m^2}{n^6}
\][/tex]
Now combine the terms:
[tex]\[
4 m n \cdot m^2 \cdot \frac{1}{n^6} = 4 m^{1+2} n^{1-6} = 4 m^3 n^{-5}
\][/tex]
Next, we need to raise this simplified expression to the power of [tex]\(-2\)[/tex]:
[tex]\[
\left(4 m^3 n^{-5}\right)^{-2}
\][/tex]
Raising a product to a power means raising each factor to that power:
[tex]\[
4^{-2} \cdot (m^3)^{-2} \cdot (n^{-5})^{-2}
\][/tex]
Calculate each term:
[tex]\[
4^{-2} = \frac{1}{16}, \quad (m^3)^{-2} = m^{-6}, \quad (n^{-5})^{-2} = n^{10}
\][/tex]
Combine these results:
[tex]\[
\frac{1}{16} \cdot m^{-6} \cdot n^{10} = \frac{n^{10}}{16 m^6}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\frac{n^{10}}{16 m^6}
\][/tex]
Among the given choices, this matches with the second option:
[tex]\[
\boxed{\frac{n^{10}}{16 m^6}}
\][/tex]
