Why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and irrational number is? if the rational number were 0, would it give us the same result we found in part b?

Part B seems to be missing, but I think I have enough information to be able to answer.

Let's say we had two numbers x and y. Let x be rational and y be irrational.

If x is some nonzero number, then x*y is irrational. The proof for this is a bit lengthy so I'll leave it out.

For instance,

x = 2 is rational, y = sqrt(3) is irrational, x*y = 2*sqrt(3) is irrational.

If we made x = 0, then

x*y = 0*y = 0

This is true for any value of y that we want. The y value doesnt even have to be irrational. It can be any real number.

So the distinction is that if x = 0, then x*y = 0 is rational since 0 is rational. Otherwise, x*y is irrational.