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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 00 is undefined.



Supplementary definitions


  Zero as an exponent
Notice that 31 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 n − m 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 x0.

Therefore we define 30 = 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.

This article is licensed under the GNU Free Documentation License. It uses material from the article "Exponentiation#Exponents one and zero" Retrieved 19 Jan 2009


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