The problem asks for the value of $2^{-5} \times 5^{-5}$. We can simplify this step by step:

1. First, let's rewrite the expression: $2^{-5} \times 5^{-5} = (2 \times 5)^{-5}$ Since both terms are raised to the power of $-5$, we can combine them under the same exponent: $(2 \times 5)^{-5} = 10^{-5}$

2. So, the value of $2^{-5} \times 5^{-5}$ is $10^{-5}$.

Thus, the correct answer is: $\boxed{C. \ 10^{-5}}$

Do you have any questions or would you like more details?

Here are 5 related questions to explore:

1. What are the general rules for multiplying numbers with negative exponents?
2. How would you simplify $(a^{-m} \times b^{-n})$ in terms of exponents?
3. What happens if you have a product of more than two terms with negative exponents?
4. How do negative exponents relate to the concept of reciprocal values?
5. How do you simplify expressions with mixed positive and negative exponents?

Tip: When multiplying terms with the same exponent but different bases, you can combine the bases first and then apply the exponent.