Simplify the boolean expression to two literals

Question

Expression: $$[AB'(C+BD) + A'B']C$$ I start off using the distributive law, and then nowhere to go.

I need help.

Answer 1

Indeed, we do need the distributive law multiple times. It's used in the first line, again in the second line, and once more in the fourth line.

$$\begin{align} [AB'(C+BD) + A'B']C & = [AB'C + A\underbrace{B'B}_{=\; 0}D+ A'B']C\\ \\ & = AB'\underbrace{CC}_{=\;C} + 0 + A'B'C \\ \\ & = AB'C + A'B'C \\ \\ & = [\underbrace{A + A'}_{=\;1}](B'C) \\ \\ & = B'C\end{align}$$