Zero as an Exponent

Date: 2/1/96 at 19:11:57
From: Anonymous
Subject: powers of zero

Dear Dr. Math, 

We know that 2 to the power of zero is one, but we have a problem
understanding some of the reasoning.  Two to the power of 3 is 2 
times itself 3 times or 2x2x2.  But, two to the power of 0 isn't 2 
times itself 0 times, is it?  Can you explain 2^0 in those terms?

Thanks,

Kathryn Gerleman
Jordan Middle School
7th grade pre algebra

Date: 3/12/96 at 1:24:37
From: Doctor Jodi
Subject: Re: powers of zero

Hi Katryn!  This is a really good question!

I'm not sure if you're familiar with negative exponents, and they always 
confuse me a bit, so let me start off with Ken's explanation of negative 
exponents.

       Negative exponents mean that instead of multiplying that many of
       the base together, you divide.  For instance, 3^2 = 9, and 3^-2 = 
       1/9.  That's one way to see why anything to the zero power 
       (except perhaps 0) is 1.  The way most people think of negative 
       exponents is "put it in the bottom of the fraction.

So, let's look at a series:

3^1 = 3 3^2 = 9 3^3 = 27, etc.

On the other side, we can add some negative exponents:

3^(-1) = 1/3 3^(-2) = 1/9 3^(-3) = 1/27, etc.

So we have two series that look like

... 1/27, 1/9, 1/3

and

3, 9, 27....

(The dots mean that you could continue the series in that direction if 
you wanted to ... for as long as you wanted.)

Do you see any pattern?

Well, in each series, each time you jump to the right, you multiply 
by 3.

But do you also notice that 1/3 and 3 have a relationship of 
multiplying/dividing by 9 (depending which way you're going)?

Well, if we add 1 as a sort of mathematical glue between the two series, 
we get

... 1/27, 1/9, 1/3, 1, 3, 9, 27

Which I think is much prettier than the two series above.

In order to make this series, we need 1... which corresponds to 3^0.

There are other explanations, too, that deal more with adding and 
subtracting exponents.  If you want to know more, or if you have other 
questions, write us back!
 
-Doctor Jodi,  The Math Forum