https://file.aiquickdraw.com/mathbot/file/1729038745043t4wcb21p.jpg

The expression shown in the image is:

\[
\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^2} - \frac{1}{b^2}}
\]

Let me break it down step by step.

### Numerator Simplification:
The numerator is:

\[
\frac{1}{a} + \frac{1}{b}
\]

To add these two fractions, find the common denominator:

\[
\frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab}
\]

So, the numerator simplifies to:

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

### Denominator Simplification:
The denominator is:

\[
\frac{1}{a^2} - \frac{1}{b^2}
\]

This is a difference of squares, and it can be factored as:

\[
\frac{1}{a^2} - \frac{1}{b^2} = \frac{(b^2 - a^2)}{a^2b^2}
\]

Since \( b^2 - a^2 = (b - a)(b + a) \), the denominator becomes:

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

### Full Expression:
Now substitute the simplified numerator and denominator back into the full expression:

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

### Final Simplification:
Dividing by a fraction is the same as multiplying by its reciprocal, so we have:

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

The \(a + b\) terms cancel out:

\[
\frac{1}{ab} \times \frac{a^2b^2}{b - a} = \frac{a b}{b - a}
\]

Thus, the simplified expression is:

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

---

Would you like more details or further clarifications? Here are some related questions you could explore:

1. How does the difference of squares apply to the denominator?
2. What happens if \(a = b\)? Does the expression become undefined?
3. Can this result be generalized for similar types of rational expressions?
4. How would the result change if the numerator involved different exponents or terms?
5. How could this expression be used in solving real-world problems involving rates?

**Tip:** Always look for factoring opportunities when simplifying complex algebraic expressions. It can often make the solution process much more efficient.

Simplify the expression \(\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^2} - \frac{1}{b^2}}\)

Learn how to simplify the expression \(\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^2} - \frac{1}{b^2}}\) using algebraic methods such as finding common denominators and applying the difference of squares. Suitable for students in Grades 9-12.

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