Simplify $\bar{a}\bar{b}(\bar{a} + b)(\bar{b} + b)$

Question

I've managed to simplify $\bar{a}\bar{b}(\bar{a} + b)(\bar{b} + b)$ to $\bar{a} +\bar{a}\bar{b}$

Where $\bar{a}$ = $a$ complement I simplified the original expression with Huntingtons Postulates, Idempotent laws etc

I've checked they are equivalent by truth table. Can this be simplified even further?

Answer 1

Recall that $xx = x$, $\bar{x} + x = 1$, and $\bar{x}x = 0$ for all $x$.

\begin{align} \bar{a}\bar{b}(\bar{a} + b)(\bar{b} + b) &= \bar{a}\bar{b}(\bar{a}+b) \\ &= \bar{a}\bar{b}\bar{a} + \bar{a}\bar{b}b \\ &= \bar{a}\bar{a}\bar{b} \\ &= \bar{a}\bar{b}. \end{align}

Line 1: $\bar{x}+x = 1$,
Line 2: $x(y+z) = xy + xz$,
Line 3: $\bar{x}x=0$,
Line 4: $xx = x$.