Final answer:
The simplified Boolean expression is AB + AC. This is derived using the idempotent, distributive, associative, commutative, and absorption laws of Boolean algebra.
Step-by-step explanation:
The question asks to simplify the Boolean expression X = (ABC)(A+C)(A+C). Using the properties of Boolean algebra, we can simplify this expression. Specifically, we apply the idempotent law, which states that A+A = A for any Boolean variable A, to the repeated (A+C) term. We also use the distributive law which in Boolean algebra can be expressed as A(B+C) = AB + AC.
The expression thus simplifies as follows:
(ABC)(A+C) (after applying idempotent law to (A+C)(A+C))
= ABCA + ABCC (distributive law)= A(BCA) + C(ABC) (associative and commutative laws)= AA(BC) + CC(AB) (associative law)= A(BC) + C(AB) (idempotent law as AA=A and CC=C)= AB + AC (absorption law where A(BC) reduce to AB and C(AB) to AC)
The simplified expression is AB + AC.