The problem requires simplifying the expression [tex]\(\left(\frac{8^{-5}}{2^{-2}}\right)^{-4}\)[/tex].
Let's break it down step-by-step.
### Step 1: Simplify the Inner Expression
First, let's focus on the expression inside the parentheses:
[tex]\[
\frac{8^{-5}}{2^{-2}}
\][/tex]
Notice that [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]:
[tex]\[
8^{-5} = (2^3)^{-5} = 2^{-15}
\][/tex]
Thus, the inner expression becomes:
[tex]\[
\frac{2^{-15}}{2^{-2}}
\][/tex]
### Step 2: Simplify the Division of Exponents
When dividing powers with the same base, we subtract the exponents:
[tex]\[
2^{-15} / 2^{-2} = 2^{-15 - (-2)} = 2^{-15 + 2} = 2^{-13}
\][/tex]
### Step 3: Apply the Outer Exponent
Now, we need to raise the simplified inner expression to the power of [tex]\(-4\)[/tex]:
[tex]\[
(2^{-13})^{-4}
\][/tex]
When raising a power to another power, we multiply the exponents:
[tex]\[
2^{-13 \cdot (-4)} = 2^{52}
\][/tex]
### Step 4: Compare to Given Options
Our expression simplifies to [tex]\(2^{52}\)[/tex].
Now let's compare this with the given options.
#### Option (A)
[tex]\[
\frac{1}{8 \cdot 2^2} = \frac{1}{8 \cdot 4} = \frac{1}{32}
\][/tex]
Clearly, [tex]\(\frac{1}{32}\)[/tex] is not equal to [tex]\(2^{52}\)[/tex].
#### Option (B)
[tex]\[
\frac{2^6}{8^9}
\][/tex]
Let's convert [tex]\(8^9\)[/tex] to base 2:
[tex]\[
8^9 = (2^3)^9 = 2^{27}
\][/tex]
Thus,
[tex]\[
\frac{2^6}{2^{27}} = 2^{6-27} = 2^{-21}
\][/tex]
Clearly, [tex]\(2^{-21}\)[/tex] is not equal to [tex]\(2^{52}\)[/tex].
#### Option (C)
[tex]\[
\frac{8^{20}}{2^8}
\][/tex]
Convert [tex]\(8^{20}\)[/tex] to base 2:
[tex]\[
8^{20} = (2^3)^{20} = 2^{60}
\][/tex]
So,
[tex]\[
\frac{2^{60}}{2^8} = 2^{60-8} = 2^{52}
\][/tex]
This matches our simplified result [tex]\(2^{52}\)[/tex].
### Conclusion
The correct option is:
[tex]\[
\boxed{\text{C} \ \frac{8^{20}}{2^8}}
\][/tex]
