The given expression is $6^{-1} \times 6^{-1}$. Let's break this down:

1. $6^{-1}$ means the reciprocal of 6, which is $\frac{1}{6}$.
2. So, the expression becomes $\frac{1}{6} \times \frac{1}{6}$.
3. Multiplying fractions: $\frac{1}{6} \times \frac{1}{6} = \frac{1 \times 1}{6 \times 6} = \frac{1}{36}$.

Thus, the correct answer is $\frac{1}{36}$.

Would you like any further details or clarifications?

Here are some related questions for deeper understanding:

1. What happens if you multiply negative exponents like $6^{-1} \times 6^{-2}$?
2. How does exponentiation work with different bases?
3. What is the general rule for multiplying powers with the same base?
4. Can negative exponents result in larger values?
5. How would $6^1 \times 6^{-1}$ simplify?

Tip: When multiplying terms with the same base and exponents, simply add the exponents (e.g., $a^m \times a^n = a^{m+n}$).