Associated Topics || Dr. Math Home || Search Dr. Math

### Zero as an Exponent

```
Date: Sat, 22 Jun 1996 11:30:52 -0400 (EDT)
From: Anonymous
Subject: Zero as an exponent

My 7th grade son has a 5^0 question on a study guide and didn't
know the answer. I told him that I thought that the answer was 1
based on my math training in years gone by, but I didn't know
why. I searched the net and found your page which explained it
well enough for me to understand.

I went on to explain it to him in this way:

5^1=(5*1)/1, 5^2 = (5*5*1)/1, 5^(-1)=1/(5*1), 5^(-2)=1/(5*5*1).

Following this flow, 5^0 would be viewed as 1/1 with no 5's. Then
of course 1/1 = 1 or 5^0 = 1.

Does this make sense? Have I got it figured correctly?

- Rick Humphreys
```

```
Date: Sat, 22 Jun 1996 17:19:00 -0400 (EDT)
From: Dr. Ceeks
Subject: Re: Zero as an exponent

Hi,

arise out of any natural sequence of ideas.

I think this is more natural:

First, the exponential was defined as a notational method to
represent the process of multiplying a given number over and over.

Thus, 5^n = 5 times 5 times 5 times 5, n times, where n is a
positive integer. It then follows that 5^(a+b) = 5^a 5^b.

In mathematics, it often happens that one would like to extend the
definition of something. How can we extend the definition of the
exponent to all the integers? What property of the exponential can
guide us beyond the positive integers?

We have the beautiful law that 5^(a+b) = 5^a 5^b. Is it possible to
extend the definition so as to retain the fundamental property that
5^(a+b) = 5^a 5^b?

The answer is yes...it can be extended, and it can be extended in
only one way.

First, 5^(-1) must be 1/5, because we demand that 5^(n-1)=5^n 5^(-
1). But then we see that 5^0 = 5^(1-1) = 5^1 5^(-1) = 5 * 1/5 = 1.

-Doctor Ceeks, The Math Forum

```
Associated Topics:
Middle School Exponents

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search