To determine which expressions are equivalent to [tex]\(\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}\)[/tex], we need to simplify each given option and then compare them to the original expression.
1. Let's simplify the original expression:
[tex]\[
\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}
\][/tex]
This can be rewritten as:
[tex]\[
\left(\frac{1}{5}\right)^4 = 5^{-4}
\][/tex]
2. Now let’s evaluate each option one by one to find out which are equivalent to [tex]\(5^{-4}\)[/tex]:
Option A: [tex]\(\left(5^{-2}\right)^2\)[/tex]
[tex]\[
(5^{-2})^2 = 5^{-2 \cdot 2} = 5^{-4}
\][/tex]
Thus, [tex]\(\left(5^{-2}\right)^2 = 5^{-4}\)[/tex], which is correct.
Option B: [tex]\(\left(5^{-4}\right)^0\)[/tex]
[tex]\[
(5^{-4})^0 = 5^{0} = 1
\][/tex]
This is not equivalent to [tex]\(5^{-4}\)[/tex], so this option is incorrect.
Option C: [tex]\(\frac{5^1}{5^4}\)[/tex]
[tex]\[
\frac{5^1}{5^4} = 5^{1-4} = 5^{-3}
\][/tex]
This is not equivalent to [tex]\(5^{-4}\)[/tex], so this option is incorrect.
Option D: [tex]\(5^2 \cdot 5^{-6}\)[/tex]
[tex]\[
5^2 \cdot 5^{-6} = 5^{2 + (-6)} = 5^{-4}
\][/tex]
Thus, [tex]\(5^2 \cdot 5^{-6} = 5^{-4}\)[/tex], which is correct.
Hence, the expressions equivalent to [tex]\(\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}\)[/tex] are:
- [tex]\(\left(5^{-2}\right)^2\)[/tex]
- [tex]\(5^2 \cdot 5^{-6}\)[/tex]
So, the two correct options are:
- (A) [tex]\(\left(5^{-2}\right)^2\)[/tex]
- (D) [tex]\(5^2 \cdot 5^{-6}\)[/tex]
