To solve the expression [tex]\( \left(\frac{8^{-5}}{2^{-2}}\right)^{-4} \)[/tex], we need to simplify it step by step.
1. Begin by simplifying the expression inside the parentheses: [tex]\(\frac{8^{-5}}{2^{-2}}\)[/tex].
2. Use the property of exponents that [tex]\(\frac{a^m}{b^n} = a^m \cdot b^{-n}\)[/tex]:
[tex]\[
\frac{8^{-5}}{2^{-2}} = 8^{-5} \cdot 2^2
\][/tex]
3. Note that [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]:
[tex]\[
8 = 2^3
\][/tex]
4. Thus, [tex]\(8^{-5}\)[/tex] can be rewritten using the base 2:
[tex]\[
8^{-5} = (2^3)^{-5} = 2^{3 \cdot (-5)} = 2^{-15}
\][/tex]
5. Substitute [tex]\(2^{-15}\)[/tex] back into the expression:
[tex]\[
2^{-15} \cdot 2^2
\][/tex]
6. Combine the exponents using the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
2^{-15 + 2} = 2^{-13}
\][/tex]
7. Now, the original expression inside the parentheses is simplified to:
[tex]\[
(2^{-13})^{-4}
\][/tex]
8. Apply the power-of-a-power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(2^{-13})^{-4} = 2^{-13 \cdot -4} = 2^{52}
\][/tex]
So, the equivalent expression for [tex]\( \left(\frac{8^{-5}}{2^{-2}}\right)^{-4} \)[/tex] is:
[tex]\[ 2^{52} \][/tex]
Finally, let's compare this result with the choices provided:
(A) [tex]\( \frac{1}{8 \cdot 2^2} \)[/tex]
(B) [tex]\( \frac{2^6}{8^9} \)[/tex]
(C) [tex]\( \frac{8^{20}}{2^8} \)[/tex]
To determine which choice matches [tex]\(2^{52}\)[/tex], let's simplify each option:
- (A) [tex]\( \frac{1}{8 \cdot 2^2} \)[/tex]
[tex]\[
8 = 2^3 \Rightarrow 8 \cdot 2^2 = 2^3 \cdot 2^2 = 2^{5}
\][/tex]
[tex]\[
\frac{1}{2^{5}} \neq 2^{52}
\][/tex]
- (B) [tex]\( \frac{2^6}{8^9} \)[/tex]
[tex]\[
8 = 2^3 \Rightarrow 8^9 = (2^3)^9 = 2^{27}
\][/tex]
[tex]\[
\frac{2^6}{2^{27}} = 2^{6-27} = 2^{-21} \neq 2^{52}
\][/tex]
- (C) [tex]\( \frac{8^{20}}{2^8} \)[/tex]
[tex]\[
8 = 2^3 \Rightarrow 8^{20} = (2^3)^{20} = 2^{60}
\][/tex]
[tex]\[
\frac{2^{60}}{2^{8}} = 2^{60-8} = 2^{52}
\][/tex]
This matches our original result of [tex]\(2^{52}\)[/tex].
Therefore, the correct choice is:
(C) [tex]\(\frac{8^{20}}{2^8}\)[/tex]
