[jkent]

I successfully managed to explain 3^0 to 10 year olds (as a recovering high school math teacher) as:

  3^2 = 9
  3^1 = 3  (divide 9 by 3)
  3^0 = 1  (divide 3 by 3)
  3^-1 = 1/3  (divide 1 by 3)
  etc

This can logically be explained as n^0=1 for all real numbers.

Unfortunately this doesn't really handle 0^0 but fortunately 10 year olds are rarely that difficult.

[amalcon]

This explanation works pretty well on adults: Just include the multiplicative identity (1) in the expansion.

  3^3 = 3*3*3*1 = 27
  3^2 = 3*3*1   = 9
  3^1 = 3*1     = 3
  3^0 = 1       = 1

and likewise:

  0^3 = 0*0*0*1 = 0
  0^2 = 0*0*1   = 0
  0^1 = 0*1     = 0
  0^0 = 1       = 1

I haven't yet tried this on an actual 10 year old, though.

[jkent]

Also good, but the last line doesn't work (in my opinion): to get from 0x1 to 1 you need to undo the x0, i.e. dividing by 0. Which is undefined ... Zero, confusing 10 year olds for centuries!

[amalcon]

True. If you approach it from a standpoint of extrapolation from known values, you have a division by zero one way or the other.

What I'd intended was that the value of 1 was reached by applying the same algorithm that was applied to arrive at the other values: start with 1, multiply by the base once per instance of the exponent. No division involved.

It's still incorrect if we want to be strict, of course. That algorithm is not quite the definition of exponentiation, because that algorithm can't really be extended to work outside rational exponents. Exponentiation is defined across complex numbers (ignoring 0^0 for the moment). I think this is acceptable because I'm only shooting for an explanation, which doesn't need to be strict.

[wlievens]

That's what the parent poster meant with "it's necessary to make the algebra of powers work out".

[pshapiro]

Thanks, I was gonna say the same thing. :)