The two expressions that need to be compared are:

1. $\frac{1}{a^5b^2}$
2. $-a^{-5}b^{-2}$

Step 1: Simplifying the second expression

We start by simplifying the second expression, $-a^{-5}b^{-2}$.

• Applying the rule of negative exponents: $x^{-n} = \frac{1}{x^n}$

This leads to: $-a^{-5}b^{-2} = -\frac{1}{a^5b^2}$

Step 2: Comparison

• The first expression is $\frac{1}{a^5b^2}$, and the simplified version of the second expression is $-\frac{1}{a^5b^2}$.

The two expressions are not equivalent because the second expression has a negative sign in front, while the first does not.

Conclusion:

The pair of expressions is not equivalent.

Would you like further details on any steps? Here are five related questions for practice:

1. What is the rule for handling negative exponents?
2. How can you rewrite $x^{-3}$ as a fraction?
3. What happens when both the numerator and denominator have negative exponents?
4. What does the negative sign in front of an expression indicate?
5. How can you verify if two algebraic expressions are equivalent?

Tip: Always simplify the expressions fully before comparing them to avoid missing subtle differences like negative signs.