We are asked to find the equivalent expression of $\left(\frac{8^{-5}}{2^{-2}}\right)^{-4}$.

1. Problem Description

We are asked to find the equivalent expression of $\left(\frac{8^{-5}}{2^{-2}}\right)^{-4}$.

2. Solution Steps

We have the expression $\left(\frac{8^{-5}}{2^{-2}}\right)^{-4}$.

First, simplify the expression inside the parenthesis.

We know that $8 = 2^3$, so we can write $8^{-5} = (2^3)^{-5} = 2^{-15}$.

So, the expression becomes $\left(\frac{2^{-15}}{2^{-2}}\right)^{-4}$.

Now, we can use the rule $\frac{a^m}{a^n} = a^{m-n}$.

Thus, we have $\frac{2^{-15}}{2^{-2}} = 2^{-15 - (-2)} = 2^{-15 + 2} = 2^{-13}$.

Then, the expression becomes $\left(2^{-13}\right)^{-4}$.

Using the rule $(a^m)^n = a^{m \times n}$, we get $2^{-13 \times -4} = 2^{52}$.

Now we check each of the answer choices.

A) $\frac{1}{8 \cdot 2^2} = \frac{1}{2^3 \cdot 2^2} = \frac{1}{2^5} = 2^{-5}$. This is not equal to $2^{52}$.

B) $\frac{2^6}{8^9} = \frac{2^6}{(2^3)^9} = \frac{2^6}{2^{27}} = 2^{6-27} = 2^{-21}$. This is not equal to $2^{52}$.

C) $\frac{8^{20}}{2^8} = \frac{(2^3)^{20}}{2^8} = \frac{2^{60}}{2^8} = 2^{60-8} = 2^{52}$.

3. Final Answer

The equivalent expression is $\frac{8^{20}}{2^8}$.

The final answer is (C).