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

### Numbers Raised to the Negative Power

```
Date: 11/14/2001 at 10:35:33
From: Rosemary
Subject: Numbers raised to the negative power

I am in 7th grade and in Pre-Algebra. We are using numbers raised to
the negative power. I know that 5^(-N) = 1/5^N. I would like to know
why. My teacher said it was by definition, but I know there is a
reason behind this definition. Any help would be appreciated.

Rosemary
```

```
Date: 11/14/2001 at 12:18:53
From: Doctor Peterson
Subject: Re: Numbers raised to the negative power

Hi, Rosemary.

Your teacher is right that negative exponents are defined this way;
but the definition is made that way for a good reason, namely so that
the properties of exponents continue to be true.

Look at this expression:

a^b * a^-b

If the property

a^b * a^c = a^(b+c)

still holds for negative exponents, then we find that

a^b * a^-b = a^(b + -b) = a^0

Therefore, our definition has to say that

a^-b = a^0 / a^b

Now, how do we define a^0? Again, consider

a^1 * a^0 = a^(1+0) = a^1

This means that

a^0 = a^1 / a^1 = 1

Now we know that we have to define a^0 as 1, and that we have to
define a^-b as 1/a^b.

See the Dr. Math FAQ for more ideas in this area (mostly about a^0):

n^0 (any number to zero power)
http://mathforum.org/dr.math/faq/faq.number.to.0power.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Number Theory

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