Problem $4)$ Boolean Expression Simplification $(40$ pts$)$

$x,Y,$ and $Z$ refer to three Boolean variables whose value can be $0$ or $1.$ Simplify given Boolean

expressions using axioms and theorems of Boolean algebra. The simplified expressions need to

be in a "Sum$-$of$-$product" format, and further simplification $($i$.$e$.,$ reducing the number of terms

in the "Sum$-$of$-$product" format$)$ shouldn't be available anymore. Show each step with a brief

description of what you did to reach the next step, as the example below shows:

Example$)$

AB'C$+$ABC

$=$ACB'$+$ACB $(commutativity)$

$=AC(B\text{'}+B)(distributivity)$

$=AC(complementarity)$

$(1)x\text{'}Y*(Y+Zx+Z\text{'}x\text{'})$

$(2)(x+Y)*(Y+Z)*(x\text{'}+Z)$