To simplify the expression \(\frac{5^{-2}}{5^8}\), we can use the properties of exponents. According to the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\), we can rewrite the expression as follows:
\[ \frac{5^{-2}}{5^8} = 5^{-2 - 8} = 5^{-10} \]
Now, we can express \(5^{-10}\) as:
\[ 5^{-10} = \frac{1}{5^{10}} \]
So, the equivalent expression for \(\frac{5^{-2}}{5^8}\) is \(\frac{1}{5^{10}}\).
Among the multiple-choice answers given:
- \(5^6\)
- \(5^{45}\)
- \(\frac{1}{5^{10}}\)
- \(\frac{1}{5^{4}}\)
- \(1\)
The correct answer is:
\[ \frac{1}{5^{10}} \]