# Advanced Math Which one is a simplified expression for A(A'+B')(B+C)(B+C'+D)

Question
Upper Level Math
Which one is a simplified expression for A(A'+B')(B+C)(B+C'+D)

2021-02-06
Step 1
Multiply A with the terms in the first bracket:
$$\displaystyle={\left[{A}{\left({A}'+{B}'\right)}\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}$$
$$\displaystyle={\left[\forall'+{A}{B}'\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}$$
Since PP'=0
$$\displaystyle={\left[{0}+{A}{B}'\right]}{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}$$
$$\displaystyle={A}{B}'{\left({B}+{C}\right)}{\left({B}+{C}'+{D}\right)}$$
Step 2
Multiply AB' with the terms in the first bracket of the obtained expression:
$$\displaystyle={\left[{A}{B}'{\left({B}+{C}\right)}\right]}{\left({B}+{C}'+{D}\right)}$$
$$\displaystyle={\left[{A}{B}{B}'+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}$$
$$\displaystyle={\left[{A}{\left({B}{B}'\right)}+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}$$
Since PP'=0
$$\displaystyle={\left[{0}+{A}{B}'{C}\right]}{\left({B}+{C}'+{D}\right)}$$
$$\displaystyle={A}{B}'{C}{\left({B}+{C}'+{D}\right)}$$
Step 3
Multiply AB'C with all the terms in the bracket of the obtained expression:
=AB'C(B+C'+D)
=ABB'C+AB'CC'+AB'CD
=A(BB')C+AB'(CC')+AB'CD
Since PP'=0
=0+0+AB'CD
=AB'CD
Thus the simplified expression is AB'CD.

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