Let's break down the given Boolean expression step by step, applying Boolean algebra rules.

Boolean Expression:

$A'(a + b) + (b + aa)(a + b')$

1. First part: $A'(a + b)$

   • Distribute $A'$ over the sum $(a + b)$: $A'a + A'b$

2. Second part: $(b + aa)(a + b')$

   • Simplify $aa$. In Boolean algebra, $aa = a$, so: $(b + a)(a + b')$
   • Now, distribute over the two terms: $ba + bb' + aa + ab'$
   • Using the complement rule $bb' = 0$: $ba + 0 + a + ab'$
   • Simplifying gives: $ba + a + ab'$

3. Final step: Combine both parts: $A'a + A'b + (ba + a + ab')$

   • Since $a + a = a$ (idempotent law), we simplify further: $A'a + A'b + ba + ab'$

This is the simplified Boolean expression: $A'a + A'b + ba + ab'$

Let me know if you'd like more details on the steps or have any specific questions!

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Here are 5 related questions to expand your understanding:

1. What is the idempotent law in Boolean algebra?
2. How does the complement rule work in Boolean algebra?
3. Can you explain the distribution property in Boolean expressions?
4. What is the significance of the absorption law in simplifying expressions?
5. How would the expression change if $A = 1$?

Tip: Remember that simplifying Boolean expressions often involves applying rules multiple times, so recognizing patterns is key!