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To solve the exercise, we need to find the coefficient of \( ab \) after simplifying the given algebraic expression. The expression is:

\[
\left( 3a^2b - \frac{ab}{2} \right) - \left( -4a^2b + \frac{2ab}{3} \right) + \left( a^2b + ab \right)
\]

Let's go step by step.

### Step 1: Expand and simplify
First, we can rewrite the expression by removing the parentheses and distributing the negative signs:

\[
= \left( 3a^2b - \frac{ab}{2} \right) - \left( -4a^2b + \frac{2ab}{3} \right) + \left( a^2b + ab \right)
\]

\[
= 3a^2b - \frac{ab}{2} + 4a^2b - \frac{2ab}{3} + a^2b + ab
\]

### Step 2: Combine like terms

#### For the \( a^2b \) terms:
\[
3a^2b + 4a^2b + a^2b = 8a^2b
\]

#### For the \( ab \) terms:
\[
-\frac{ab}{2} - \frac{2ab}{3} + ab
\]

We need to combine the fractions. The least common denominator for 2 and 3 is 6:

\[
-\frac{ab}{2} = -\frac{3ab}{6}, \quad -\frac{2ab}{3} = -\frac{4ab}{6}, \quad ab = \frac{6ab}{6}
\]

Now, add these fractions:

\[
-\frac{3ab}{6} - \frac{4ab}{6} + \frac{6ab}{6} = \frac{-3 - 4 + 6}{6}ab = \frac{-1}{6}ab
\]

### Step 3: Final expression
The simplified expression becomes:

\[
8a^2b - \frac{1}{6}ab
\]

### Step 4: Coefficient of \( ab \)
The coefficient of \( ab \) is \( -\frac{1}{6} \).

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Let me know if you'd like further details or have questions!

Here are five related questions:
1. How do you simplify algebraic expressions with fractions?
2. What is the method for finding the least common denominator?
3. How do you combine like terms in algebraic expressions?
4. How can we factor expressions involving similar terms?
5. How does distributing negative signs affect the terms in an expression?

**Tip**: When combining fractions with different denominators, always look for the least common denominator to simplify calculations.

Quel est le coefficient de «ab» après avoir réduit l'expression algébrique suivante?

Learn how to find the coefficient of 'ab' in an algebraic expression by simplifying the expression step-by-step. This problem involves key algebraic concepts like combining like terms and working with fractions, suitable for students in Grades 9-10.

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