Number theory

Question

Is any number powered by 0 a 1? Including 0 itself? Thanks

Answer

Brett, Let's look at "a to the power b", which we'll write as a^b. When b is a positive integer, this just means a*a*a*a*...*a, where there are b a's in total (and I'm using * for multiplication). It's quite easy to see then that a^(b+c) = (a^b)*(a^c), since both sides have (b+c) a's multiplied together. So what should a^b be when b is zero? Well, if we want the above formula to still work, we get a^c = a^(0+c) = (a^0) * (a^c). Dividing by a^c, we see that (a^0) must be 1. But this argument doesn't work for a = 0, because we'd have to divide by 0. Just as we can't define 1/0 without getting into trouble, we can't define 0^0. You might ask yourself what should a^(-b) be? You should be able to use a similar argument to find out.