In boolean algebra, why is a+a'b = a+b?

Question

Looking at the truth table, they're equivalent:

A   B      (A+B)  (A+A'B)
-------------------------------
1   1       1       1
1   0       1       1
0   1       1       1
0   0       0       0

But what manipulation can one do using basic identifies and laws to show that they're the same?

Answer 1

Note with the laws of Boolean algebra, "addition" distributes over "multiplication" (just as multiplication would normally distribute over addition). Thus, we have $$ a + (a'\cdot b) = (a+a')\cdot (a+b) = 1(a+b) = a+b $$

Answer 2

$A+A'B=A(1+B)+A'B=A+AB+A'B=A+(A+A')B=A+B$