[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!

[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. :)