"Here is as good a place as any to introduce into our logical discussion of the real number system the notion of exponentiation with integral exponents. If a [is a real number] and n is some positive integer, we define to be a·a·a···a (n times), and if a [is not equal to zero], , . From these definitions we immediately derive the usual rules of exponentiation, in particular , , [and] ." (see ita-2-2-16-pg21)
The 'usual rules of exponentiation' are evident from the definition of exponentiation where the exponent, n, is positive. The rationale when the exponent is zero or negative requires calculus. However, given that we have chosen as the symbol for the multiplicative inverse, we can note the following:
Below are some examples of exponentiation in Mathematica.