Let's simplify the expression:

\[
\frac{x^{-2} - y^{-2}}{x^{-1} - y^{-1}}
\]

### Step 1: Simplify the exponents
Recall that \( x^{-n} = \frac{1}{x^n} \). So, we can rewrite the expression as:

\[
\frac{\frac{1}{x^2} - \frac{1}{y^2}}{\frac{1}{x} - \frac{1}{y}}
\]

### Step 2: Simplify the numerator
The numerator \( \frac{1}{x^2} - \frac{1}{y^2} \) can be rewritten using the difference of squares formula:

\[
\frac{1}{x^2} - \frac{1}{y^2} = \frac{y^2 - x^2}{x^2y^2} = \frac{(y-x)(y+x)}{x^2y^2}
\]

### Step 3: Simplify the denominator
The denominator \( \frac{1}{x} - \frac{1}{y} \) can be written as:

\[
\frac{1}{x} - \frac{1}{y} = \frac{y-x}{xy}
\]

### Step 4: Combine the numerator and denominator
Now, the entire expression is:

\[
\frac{\frac{(y-x)(y+x)}{x^2y^2}}{\frac{y-x}{xy}}
\]

This can be simplified by multiplying the numerator by the reciprocal of the denominator:

\[
\frac{(y-x)(y+x)}{x^2y^2} \times \frac{xy}{y-x}
\]

Notice that \( (y-x) \) cancels out:

\[
\frac{y+x}{xy}
\]

### Final Answer:
The simplified expression is:

\[
\frac{y+x}{xy}
\]

Do you have any questions or need further explanation?

Here are some related questions you might consider:
1. How would the expression change if the exponents were positive?
2. How can you simplify similar expressions with higher negative exponents?
3. What if the expression involved three variables instead of two?
4. How does this simplification process change for fractional exponents?
5. Can you factor the expression differently and arrive at the same result?
6. What if the expression was in terms of addition instead of subtraction?
7. How does the expression change if \( x = y \)?
8. How do you simplify complex expressions involving both positive and negative exponents?

**Tip:** When dealing with negative exponents, always consider rewriting them as fractions to simplify the expression more easily.

Learn how to simplify the expression (x^-2 - y^-2) / (x^-1 - y^-1) step-by-step. This problem involves exponents and fractional expressions, using the difference of squares formula for simplification. Suitable for students in Grades 9-12.

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