https://file.aiquickdraw.com/mathbot/file/1730142364558hpnua7ft.jpg

The expression in the image appears to be as follows:

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

Let's simplify this step-by-step.

### Step 1: Simplify \(\frac{a^2}{a^2 - b^2}\)
The term \(a^2 - b^2\) can be factored using the difference of squares formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]

So, we can rewrite \(\frac{a^2}{a^2 - b^2}\) as:
\[
\frac{a^2}{a^2 - b^2} = \frac{a^2}{(a - b)(a + b)}
\]

### Step 2: Substitute and Simplify
Substitute this into the original expression:
\[
(a - b) \cdot \frac{a^2}{(a - b)(a + b)} + \frac{ab}{a - b}
\]

Now, observe that in the first term, \((a - b)\) in the numerator and denominator cancels out:
\[
\frac{a^2}{a + b} + \frac{ab}{a - b}
\]

### Step 3: Find a Common Denominator
To combine the terms, we need a common denominator, which is \((a + b)(a - b)\). Rewrite each fraction with this denominator:

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

Expanding the numerators:

1. For the first term: \(a^2(a - b) = a^3 - a^2b\)
2. For the second term: \(ab(a + b) = a^2b + ab^2\)

Now the expression becomes:
\[
\frac{a^3 - a^2b + a^2b + ab^2}{(a + b)(a - b)}
\]

### Step 4: Simplify the Numerator
Combine like terms in the numerator:
\[
a^3 + ab^2
\]

Thus, the final simplified expression is:
\[
\frac{a^3 + ab^2}{(a + b)(a - b)}
\]

Would you like a further breakdown of any step or have any questions?

Here are 5 related questions that you might find interesting:

1. How can the difference of squares formula be used in other algebraic simplifications?
2. What happens if we set \(a = b\) in this expression?
3. How would this expression change if we added a constant term, such as \(c\)?
4. How is finding a common denominator helpful when combining fractions?
5. Can this simplified expression be factored further if specific values are assigned to \(a\) and \(b\)?

**Tip:** Always check for factorization opportunities like the difference of squares to simplify complex algebraic fractions.

Simplify the expression (a - b) * (a^2 / (a^2 - b^2)) + (ab / (a - b)).

Learn how to simplify the algebraic expression (a - b)(a^2 / (a^2 - b^2)) + (ab / (a - b)) using key algebraic techniques such as the difference of squares and fraction simplification. This step-by-step solution is designed for students in Grades 9-10.

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