How to prove it (write it)?

I am not positive exactly what you are asking, but if you are trying to represent a number raised to the zero power as factors, as you have above for non-zero powers, you cannot. A number raised to the power of zero is simply defined to be equal to 1.

Hope this is what you're looking for. I know i saw the answer to this before, but forgot what it looked like, but this is how i would write it.

(-K)^2 = (-K)^1 * (-K)^1 = (-k)^(1+1) = k ^2

(-k)^3 = (-k) ^1 * (-k) ^1 * (-k) ^1 = (-k) ^ (1 + 1 + 1) = -k ^3

(-k)^0 = (-k) ^1 * (-k) ^ (-1) = (-k)/(-k) = 1

what fachatta said, only without losing the exponents (2 and 3, respectively) on the first and second examples.

(-8)/(-8) = 1
(-8)/(-8) = [(-2)^3]/[(-2)^3] = (-2)^(3-3) = (-2)^0
So, (-2)^0 = 1



I find it helpful to use numbers before I use variables. It helps prevent silly mistakes.

Cool. Thanks guys.

Yes, that is certainly correct.

Yep. your explanation made the most sense-not that I'm not grateful to the others for their input.

Lifted straight from my (memory of) calculus notes, so I can take no credit. The prof was a sadistic bastard, but his attention to basic principles was exemplary.

A lesson my chemistry professor last year still needs to learn. I even heard other academics questioning the way she imparted the information. I did well in the subject, but it was mainly due to some good sites I found on the web. Not because of the lectures.

With the best instructors, I never have to open the textbook. With the worst, I learn everything from it. Very good bacteriology prof right now -- actually uses the chalkboard instead of a computer. Forces him to make the material intelligible and sequential, instead of a mass of information.