The expression you're working with is:

$\frac{1}{5^{-3}} \cdot \frac{1}{5^6}$

To simplify this, let's break it down:

1. Using properties of exponents: We know that $\frac{1}{5^{-3}} = 5^{3}$, because the negative exponent means we take the reciprocal and flip the sign of the exponent.

2. Simplifying the multiplication: Now, we have $5^{3} \cdot 5^{-6}$.

3. Combining exponents: When multiplying terms with the same base, you add the exponents. So, this becomes:

$5^{3 + (-6)} = 5^{-3}$

4. Final simplification: $\frac{1}{5^{-3}} = 5^{3}$.

Therefore, the value of the expression is:

$5^3 = 125$

Thus, the value of the expression is 125.

Would you like further details on this explanation or have any other questions?

Here are five related questions to explore:

1. What is the rule for multiplying exponents with the same base?
2. How do you simplify expressions with negative exponents?
3. What is the reciprocal of a number with a negative exponent?
4. How do you handle expressions with both positive and negative exponents?
5. What is the difference between multiplying and dividing numbers with exponents?

Tip: Remember that $a^m \cdot a^n = a^{m+n}$, and use this rule when working with exponents!