The expression you provided is:

\[
\frac{a^{-1} - b^{-1}}{a^{-3} - b^{-3}}
\]

We can simplify this step by step.

### Step 1: Simplify the numerator and denominator using the negative exponent rule.

Recall that \( a^{-n} = \frac{1}{a^n} \), so we can rewrite both the numerator and denominator as follows:

\[
\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a^3} - \frac{1}{b^3}}
\]

### Step 2: Find a common denominator for both the numerator and denominator.

#### For the numerator:
\[
\frac{1}{a} - \frac{1}{b} = \frac{b - a}{ab}
\]

#### For the denominator:
\[
\frac{1}{a^3} - \frac{1}{b^3} = \frac{b^3 - a^3}{a^3 b^3}
\]

### Step 3: Substitute these back into the original expression.

Now, the expression becomes:

\[
\frac{\frac{b - a}{ab}}{\frac{b^3 - a^3}{a^3 b^3}}
\]

### Step 4: Simplify the complex fraction.

To simplify, multiply the numerator by the reciprocal of the denominator:

\[
= \frac{b - a}{ab} \times \frac{a^3 b^3}{b^3 - a^3}
\]

Simplifying further, the \(ab\) terms cancel out with \(a^2 b^2\) in the denominator:

\[
= \frac{(b - a) a^2 b^2}{b^3 - a^3}
\]

### Step 5: Use the difference of cubes identity.

We can factor \( b^3 - a^3 \) as \( (b - a)(b^2 + ab + a^2) \):

\[
= \frac{(b - a) a^2 b^2}{(b - a)(b^2 + ab + a^2)}
\]

### Step 6: Cancel out the common factor \( (b - a) \).

\[
= \frac{a^2 b^2}{b^2 + ab + a^2}
\]

This is the simplified form of the given expression.

### Final Answer:

\[
\frac{a^2 b^2}{b^2 + ab + a^2}
\]

Would you like further details on any part of this process?

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**Related questions:**
1. How do you simplify expressions with negative exponents?
2. Can you explain the difference of cubes formula in detail?
3. What are common techniques for simplifying complex fractions?
4. How do you identify when to factor a cubic expression?
5. What are some strategies for factoring polynomials efficiently?

**Tip:** Always look for common factors in both the numerator and denominator when simplifying rational expressions.

This detailed solution walks through simplifying the rational expression a^-1 - b^-1 / a^-3 - b^-3, applying the negative exponent rule and the difference of cubes identity. Ideal for high school students, this example covers key algebraic techniques.

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