To simplify the given expression $\frac{x^{-5}}{x^{-3}}$, you apply the quotient rule for exponents. The rule states:

$\frac{x^a}{x^b} = x^{a-b}$

Here, $a = -5$ and $b = -3$. Applying the rule:

$\frac{x^{-5}}{x^{-3}} = x^{-5 - (-3)} = x^{-5 + 3} = x^{-2}$

Thus, the simplified expression is $x^{-2}$, which corresponds to option a: $\frac{1}{x^2}$.

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

Here are 5 related questions to explore further:

1. How would you simplify $\frac{x^7}{x^2}$?
2. What is the general rule for negative exponents?
3. How does multiplying powers of the same base work?
4. How do you simplify an expression like $(x^{-3})^2$?
5. What happens when you divide numbers with exponents but different bases?

Tip: When subtracting exponents in a fraction, always remember to account for negative signs properly!