I'll stick with the grade-school math approach, at least until I need to approach it differently.

  4^2 = 2 fours multiplied = 4 * 4 = 16

divide by 4 - so you take away one of the 4s by division(canceling like terms like we do in grade school fraction math):

  4^1 = 4*4/4 = 4

divide by 4 again

  4^0 = 4*4/(4*4) = 1

divide by 4 again!

  4^-1 = 4*4/(4*4*4) = 1/4

Now try it with 0:

  0^2 = two zeros multiplied = 0*0 = 0

Divide by 0. Uh-oh. Well, let's keep following grade school fraction math and cancel like terms:

  0^1 = 0*0 / 0 = 0

Divide by 0. Hmm - keep canceling like terms.

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

But what's 0^-1? grin