Certainly! Let's simplify the given expression step by step.
We start with the expression:
[tex]\[
\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}
\][/tex]
Step 1: Simplify the inner expression:
[tex]\[
\frac{4 m n}{m^{-2} n^6}
\][/tex]
First, let's simplify the fraction inside:
[tex]\[
\frac{4 m n}{m^{-2} n^6} = 4 m n \cdot m^2 \cdot \frac{1}{n^6} = 4 m^{1+2} n^{1-6} = 4 m^3 n^{-5}
\][/tex]
So, the expression simplifies to:
[tex]\[
(4 m^3 n^{-5})^{-2}
\][/tex]
Step 2: Apply the exponent [tex]\(-2\)[/tex] to each term inside the parentheses:
[tex]\[
(4 m^3 n^{-5})^{-2} = 4^{-2} \cdot (m^3)^{-2} \cdot (n^{-5})^{-2}
\][/tex]
Step 3: Simplify each term separately:
[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 the simplified terms:
[tex]\[
(4 m^3 n^{-5})^{-2} = \frac{1}{16} \cdot m^{-6} \cdot n^{10} = \frac{n^{10}}{16 m^6}
\][/tex]
Therefore, the expression:
[tex]\[
\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}
\][/tex]
is equivalent to:
[tex]\[
\frac{n^{10}}{16 m^6}
\][/tex]
Thus, the correct option is:
[tex]\[
\frac{n^{10}}{16 m^6}
\][/tex]
