Zero as an exponent

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Definitions
 Zero as an exponent 1. Any nonzero number with an exponent of 0 is 1. The case of $0^0$ is undefined.

Supplementary definitions

Zero as an exponent
Notice that $3^1$ is the product of only one 3, which is evidently 3.

Also note that $3^5=3\cdot3^4$. Also $3^4=3\cdot{3^3}$. Continuing this trend, we should have

$3^1=3\cdot3^0$.

Another way of saying this is that when n, m, and nm are positive (and if x is not equal to zero), one can see by counting the number of occurrences of x that

$\frac{x^n}{x^m} = x^{n - m}.$

Extended to the case that n and m are equal, the equation would read

$1 = \frac{x^n}{x^n} = x^{n - n} = x^0$

since both the numerator and the denominator are equal. Therefore we take this as the definition of x$0$.

Therefore we define $3^0=1$ so that the above equality holds. This leads to the following rule:

* Any number to the power 1 is itself.
* Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products.