To solve the given expression [tex]\(\frac{25^{-4} \times 5^{-3}}{5^{-5}}\)[/tex], let's follow the steps one by one.
1. Rewrite 25 as a power of 5:
- We know that [tex]\(25 = 5^2\)[/tex]. Therefore,
[tex]\[
25^{-4} = (5^2)^{-4} = 5^{2 \times -4} = 5^{-8}
\][/tex]
2. Substitute back into the original expression:
- Now, substitute [tex]\(25^{-4}\)[/tex] with [tex]\(5^{-8}\)[/tex]:
[tex]\[
\frac{5^{-8} \times 5^{-3}}{5^{-5}}
\][/tex]
3. Combine the exponents in the numerator:
- Use the property of exponents: [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[
5^{-8} \times 5^{-3} = 5^{-8 + (-3)} = 5^{-11}
\][/tex]
- So the expression simplifies to:
[tex]\[
\frac{5^{-11}}{5^{-5}}
\][/tex]
4. Simplify the fraction using properties of exponents:
- When dividing exponents with the same base, we subtract the exponents: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{5^{-11}}{5^{-5}} = 5^{-11 - (-5)} = 5^{-11 + 5} = 5^{-6}
\][/tex]
5. Identify the simplified form:
- The simplified result is:
[tex]\[
5^{-6}
\][/tex]
Therefore, the solution to the given expression [tex]\(\frac{25^{-4} \times 5^{-3}}{5^{-5}}\)[/tex] is [tex]\(5^{-6}\)[/tex], which can be numerically evaluated as approximately [tex]\(6.4 \times 10^{-5}\)[/tex]. Among the given options, the correct answer is:
[tex]\(\boxed{5^{-6}}\)[/tex]
