Zero as an exponent

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Zero as an exponent
  • 1. Any nonzero number with an exponent of 0 is 1. The case of [math]0^0[/math] is undefined.

Supplementary definitions

Wikipedia svg logo-en.svg  Zero as an exponent
Notice that [math]3^1[/math] is the product of only one 3, which is evidently 3.

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


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

[math] \frac{x^n}{x^m} = x^{n - m}.[/math]

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

[math] 1 = \frac{x^n}{x^n} = x^{n - n} = x^0 [/math]

since both the numerator and the denominator are equal. Therefore we take this as the definition of x[math]0[/math].

Therefore we define [math]3^0=1[/math] 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.

This extract is licensed under the Creative Commons Attribution-ShareAlike license. It uses material from the article "Exponentiation#Exponents one and zero", retrieved 19 Jan 2009.