Comment by 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.