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GK proves negative exponents.
It is Catherine's theory that how our schools treat our gifted can be a pretty good measure of academic rigor in general.
I would add that we can learn a good deal about mathematical thinking, and all its various forms, from observing the gifted kids. The problem is keeping up with them. That's why my blog is called "Clueless Mom of Gifted Kid."
update from Catherine:
I asked Barry whether this is a proof - it is!
Yes, that would constitute a proof, and even though it proves it for 3 ^(-2) one can see that it extends to all numbers. A more general proof would be that since a^m/a^n = a^(m-n). If n > m, then m-n is negative. Since it's the same as dividing a^m by a^n, one can see that there are m a's in the numerator and n a's in the denominator. Through cancellation one is left with 1/a^(m-n).
There are more rigorous and formal proofs but the above is suitable for an algebra 1 course.
