To determine which expression is equivalent to [tex]\(\frac{5^8}{5^8}\)[/tex], we can analyze the given options step-by-step.
First, let's consider the original expression [tex]\(\frac{5^8}{5^8}\)[/tex]:
1. We know that when we divide any non-zero number by itself, the result is 1. Therefore:
[tex]\[
\frac{5^8}{5^8} = 1
\][/tex]
Now, we need to analyze the given options to see which one is equivalent to 1:
1. [tex]\(\frac{1}{5}\)[/tex]
[tex]\[
\frac{1}{5} = 0.2 \quad \text{(This is not equal to 1)}
\][/tex]
2. [tex]\(5^0\)[/tex]
[tex]\[
5^0 = 1 \quad \text{(Any non-zero number raised to the power of 0 is 1)}
\][/tex]
3. [tex]\(\left(5^6\right)^{-5}\)[/tex]
[tex]\[
\left(5^6\right)^{-5} = 5^{6 \cdot -5} = 5^{-30} = \frac{1}{5^{30}} \quad \text{(This is not equal to 1, but a very small positive number)}
\][/tex]
4. 5
[tex]\[
5 = 5 \quad \text{(This is not equal to 1)}
\][/tex]
Based on these evaluations, the only expression that is equivalent to [tex]\(1\)[/tex] is:
[tex]\[
5^0
\][/tex]
Therefore, the equivalent expression to [tex]\(\frac{5^8}{5^8}\)[/tex] is [tex]\(5^0\)[/tex].
