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### Anything to the 0 Power

```
Date: 11/21/97 at 16:00:46
From: Jenna M Gambino
Subject: N^0

Dear Dr. Math,

Everyone in my 7th grade class wants to know the answer to this
question: Why is anything to the 0 power 1?

No one in my class knows the answer, not even my teacher. She asked me
to e-mail Dr. Math and we are eagerly waiting to hear the answer.

Jenna
```

```
Date: 11/24/97 at 14:44:56
From: Doctor Steven
Subject: Re: N^0

Look at the powers of any number, say 8:

8^1 =    8
8^2 =   64
8^3 =  512
.
.
.

To get from 8^3 to 8^2 we have to divide by 8. And to get from 8^2 to
8^1 we have to divide by 8 again.  The logical idea is to continue
doing this so we get:

to go from 8^1 to 8^0 we should divide by 8.

So 8^0 = 8^1/8 = 8/8 = 1. This works for any nonzero number. 0^0 is
undefined.

We can even continue this process to create negative exponents:

to go from 8^0 to 8^(-1) we divide by 8

So 8^(-1) = 8^0/8 = 1/8.

Now we have this:

.
.
.
8^3    = 512
8^2    =  64
8^1    =   8
8^0    =   1
8^(-1) =   1/8
8^(-2) =   1/64
8^(-3) =   1/512
.
.
.

Hope this helps.

For more, see the Dr. Math FAQ:

http://mathforum.org/dr.math/faq/faq.number.to.0power.html

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

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