The exponential expression [tex]\(\left(\frac{1}{8}\right)^4\)[/tex] is equivalent to which of the following? Explain your choice.

A. [tex]\(4^{-8}\)[/tex]
B. [tex]\(8^{-2}\)[/tex]
C. [tex]\(2^{-12}\)[/tex]
D. [tex]\(32^{-1}\)[/tex]



Answer :

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).