That would be "intention". I am asking how you are defining "x to the n power". Definitions in mathematics are "working definitions"- you use the precise words of the definitions in proofs. Your first thought in proving anything about anything should be "what is its precise definition?"
The most common definition of xn, for n a positive integer is a "recursive" one. x1= x and xn+1= x(xn).
With that you can prove xmxn= xn+m by induction on m.
First prove: xnx1= xn+1. That follows from the definition: since x1= x, so xnx1= xxn= xn+1 from the second part of the definition.
Now suppose xnxk= xn+ k (the "inductive hypothesis"). Then xnxk+1= (xnxk)x= (xn+k)x= x(n+k)+1= xn+(k+1).
We do not "prove" that xnxm= xn+ m for m and n other than positive integers so much as we define the operation so that useful formula is true.
For example, n+0= n so in order to have [math]xnx0= xn+0= xn true, we must define x0= 1. (In order to go from xnx0= x0 to x0= 1, we must divide by xn and so must require that x not be 0. x0 is defined to be 1 for x not equal to 0 and 00 is not defined.)
n+ -n= 0 so in order to have [math]xnx-n= xn-n= x0= 1, we must define