Let's analyze the given problem step by step. The goal is to find which option the expression [tex]\(\left(\frac{1}{8}\right)^4\)[/tex] is equivalent to.
1. Starting with the given expression:
[tex]\[
\left(\frac{1}{8}\right)^4
\][/tex]
2. Rewrite [tex]\(\frac{1}{8}\)[/tex] as a power of 8:
[tex]\[
\frac{1}{8} = 8^{-1}
\][/tex]
Therefore:
[tex]\[
\left(\frac{1}{8}\right)^4 = \left(8^{-1}\right)^4
\][/tex]
3. Apply the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
\left(8^{-1}\right)^4 = 8^{-4}
\][/tex]
4. Convert 8 to a power of 2 (since [tex]\(8 = 2^3\)[/tex]):
[tex]\[
8^{-4} = (2^3)^{-4}
\][/tex]
5. Apply the power of a power rule again:
[tex]\[
(2^3)^{-4} = 2^{3 \cdot -4} = 2^{-12}
\][/tex]
6. Compare with the given options:
- (1) [tex]\(4^{-8}\)[/tex]
- (3) [tex]\(8^{-2}\)[/tex]
- (2) [tex]\(2^{-12}\)[/tex]
- (4) [tex]\(32^{-1}\)[/tex]
We can see that:
[tex]\[
2^{-12}
\][/tex]
matches our transformed expression exactly.
Hence, the exponential expression [tex]\(\left(\frac{1}{8}\right)^4\)[/tex] is equivalent to [tex]\(2^{-12}\)[/tex], which corresponds to option (2).
