The powerball lotteries I have seen don’t take the powerball from a different set of balls, it comes from the same set. It is simply one of the those drawn that is designated the powerball - usually the last.
The point is to provide a wider range of minor prizes and allow greater freedom in tuning the odds and prize values. If you have a simple 6 from 36 (or whatever number) there are only a small number of useful division, all 6, 5 and 4 numbers correct. Simply getting three number is going to be too common to allocate a prize division too. But add the power ball into the mix and you can have all 6, - which implicitly includes the powerball, 4 and powerball, 5, 3 and powerball. 4, and 2 plus powerball. Two plus the powerball is significantly harder odds than just three right, and is a minor division that can have a meaningful prize value (a few dolllars rather than a few cnets.)