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

### Zero as an Exponent

```
Date: 7/15/96 at 16:40:9
From: Anonymous
Subject: Exponent of 0

Why do you get 1 when you take a number to the 0 power?

Thanks!

Thai-Meister
```

```
Date: 7/15/96 at 17:8:27
From: Doctor Paul
Subject: Re: Exponent of 0

Let's first look at an example.  Let's look at the list of numbers
3^1, 3^2, 3^3, 3^4, ....   Finding the actual values, we get
3,   9,  27,  81, ....

So what is the pattern in the bottom sequence? Well, every time you
move to the right in the list you multiply by 3, and every time you
move to the left in the list you divide by 3. So we could take the
bottom sequence and keep going to the left and dividing by 3, and we'd
have the sequence that looks like this:

..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....
..., 1/27,  1/9,  1/3,  1,   3,   9,   27,  81, ....

So now we know what all the powers of 3 are! Actually, we just did
the integer powers of 3. But that's probably enough for now.

Now the trick is to show that you can do the same thing for any
number, not just 3. It's actually not all that hard. So once you do
that, you'll see that the x^0 term is always going to be 1, so that
means that x^0 is 1 for all x (except maybe 0^0, but that's a whole
other discussion)!

-Doctor Paul,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Number Theory
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

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/