Let's simplify the expression \( ab(a - b) - a(a - b^2) \).

### Step 1: Expand both parts of the expression

First, expand the term \( ab(a - b) \):

\[
ab(a - b) = ab \cdot a - ab \cdot b = a^2b - ab^2
\]

Next, expand the term \( a(a - b^2) \):

\[
a(a - b^2) = a \cdot a - a \cdot b^2 = a^2 - ab^2
\]

### Step 2: Subtract the second expanded expression from the first

Now, substitute these expanded forms back into the original expression:

\[
a^2b - ab^2 - (a^2 - ab^2)
\]

Distribute the negative sign across the second expression:

\[
a^2b - ab^2 - a^2 + ab^2
\]

### Step 3: Simplify the expression

Notice that the terms \( -ab^2 \) and \( +ab^2 \) cancel each other out:

\[
a^2b - a^2
\]

So, the simplified expression is:

\[
a^2b - a^2
\]

Would you like more details or have any questions?

Here are five related questions:

1. How would you factor the expression \( a^2b - a^2 \)?
2. What is the expanded form of \( a(b - c) - b(a - c) \)?
3. How would you simplify the expression \( ab(a - b) + ab(b - a) \)?
4. What happens if you substitute specific values for \( a \) and \( b \) in the expression \( ab(a - b) - a(a - b^2) \)?
5. How would you approach simplifying a more complex expression like \( ab(a - b) - a(a - b^2) + c(a - b^3) \)?

**Tip:** Always double-check the distribution of negative signs when subtracting expressions to avoid common mistakes.

Learn how to simplify the expression ab(a - b) - a(a - b^2) step-by-step with detailed explanations. This problem involves algebraic expansion and simplification techniques, suitable for advanced high school students.

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