We can simplify the Boolean expression step by step:

Given:
\[ ab + c' + c(a' + b') \]

### Step 1: Expand the third term
Use distribution to expand \( c(a' + b') \):
\[
ab + c' + ca' + cb'
\]

### Step 2: Group like terms
\[
ab + c' + ca' + cb'
\]

### Step 3: Simplify using Boolean algebra rules

- **Identity Rule**: \( x + x' = 1 \)
  In the current form, we can't simplify further unless we apply specific scenarios. Therefore, this expression can't be reduced beyond this point using fundamental Boolean algebra rules.

The simplified expression is:
\[
ab + c' + ca' + cb'
\]

Would you like further details or have any questions? Here are some related questions to practice:

1. How would you simplify \( (a + b)(a' + c) \)?
2. What is De Morgan's Law in Boolean algebra?
3. How do you factor the expression \( ab + ac + bc \)?
4. What is the distributive property in Boolean algebra?
5. How do you use Karnaugh maps to simplify Boolean expressions?

**Tip:** When simplifying Boolean expressions, always look for terms that can be combined using fundamental laws such as the distributive, absorption, or De Morgan’s laws.

This guide explains how to simplify the Boolean expression ab + c' + c(a' + b'). Learn the step-by-step process using distribution and Boolean algebra rules like the Identity Rule and De Morgan’s Law. Ideal for high school students exploring logic and circuit design.

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