Let's work through the given expression step-by-step to find its equivalent form.
We start with the expression:
[tex]\[
\left(a^{-2} \cdot 8^7\right)^2
\][/tex]
First, apply the power rule [tex]\((x^m)^n = x^{mn}\)[/tex] to each term inside the parentheses:
[tex]\[
(a^{-2})^2 \cdot (8^7)^2
\][/tex]
Now, simplify each term separately:
[tex]\[
(a^{-2})^2 = a^{-4}
\][/tex]
[tex]\[
(8^7)^2 = 8^{7 \cdot 2} = 8^{14}
\][/tex]
Combining these results, the expression simplifies to:
[tex]\[
a^{-4} \cdot 8^{14}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
\boxed{a^{-4} \cdot 8^{14}}
\][/tex]
This corresponds to option (C).
