Geoff wrote:Why is X raised to the 0 power = 1 and not 0 ?
Well, you know that fractional exponents are roots. So x^(1/2) (that is, x raised to the power of one half) is the square root of x, and x^(1/3) is the cubed root. So:
16^(1/2) = 4
16^(1/3) = 2.519842
16^(1/4) = 2
16^(1/5) = 1.741101
16^(1/6) = 1.587401
16^(1/10) = 1.319508
16^(1/15) = 1.203025
16^(1/25) = 1.117287
As the exponent gets smaller and smaller (closer to 0), the value of the expression approaches 1.
This makes intuitive sense. If the number y multipled by itself 25 times equals 16:
16^(1/25) = y --> y^25 = 16
then it is clear that y must be at least a little larger than one. The same is also true for all numbers z larger than 25.
16^(1/z) = y --> y^z = 16
As z increases into infinity, the exponent 1/z approaches 0, and the root y approaches 1.
So it makes sense to define 16^0 = 1.
You also know that negative exponents are the reciprocals, in other words, x^(-y) = 1/(x^y)
16^(-1) = 1/16 = 0.0625
16^(-2) = 1/16^2 = 0.003906
16^(-3) = 1/16^3 = 0.000244
16^(-4) = 1/16^4 = 0.000015
So as the exponent becomes more negative, the value of x^y approaches zero.
But as the exponent approaches zero from the negative direction, the value of x^y approaches one:
16^(-1/2) = 0.25
16^(-1/3) = 0.396850
16^(-1/4) = 0.5
16^(-1/5) = 0.574349
16^(-1/6) = 0.629961
16^(-1/10) = 0.757858
16^(-1/15) = 0.831238
16^(-1/25) = 0.895025
And that's another good reason to define 16^0 = 1.