Negative Exponents Explained

Date: 12/29/2001 at 16:57:12
From: Alec
Subject: Solving for x

The question is, 5 to the negative second. I have a slight problem 
with exponents and I was hoping you could shed some light on the 
subject  

Thanks.

Date: 12/29/2001 at 19:00:13
From: Doctor Achilles
Subject: Re: Solving for x

Hi Alec,

Thanks for writing to Dr. Math.

First, a bit about notation. I use the ^ symbol to designate 
exponents. So 3^4 is "three to the fourth" or a 3 with a little 4 
above and to the right. And 2^(-5) is "two to the minus fifth" or "two 
to the negative 5th."

Negative exponents are something that I think is not taught very well 
in most schools. Here's what most people are taught:

A negative exponent means you just turn the number into a fraction and 
then take the positive exponent.

So 3^(-2) is the same thing as (1/3)^2 or (1/3)*(1/3) or 1/9.

And 5^(-3) is the same thing as (1/5)^3 or (1/5)*(1/5)*(1/5) or 1/125.

This is correct, but it doesn't really explain _why_ it is correct.  
Here's how I like to think about negative exponents.

I start by thinking about positive exponents. Let's look at powers 
of 5:

  5^3 = 5*5*5 = 125

  5^2 = 5*5 = 25

  5^1 = 5 = 5

Now, how do I get from 5^3 to 5^2? Here's another way to look at it:

  5^3 = 125

  5^2 = 5^3/5 = 125/5 = 25

So 5^2 is really just 5^3 divided by 5.

  5^1 = 5^2/5 = 25/5 = 5

And 5^1 is really just 5^2 divided by 5.

and we can keep going:

  5^0 = 5^1/5 = 5/5 = 1

Aha! So _that's_ why people always say "anything to the zero power 
is 1."

  5^(-1) = 5^0/5 = 1/5

So 5^(-1) is really just 5^0 divided by 5.

  5^(-2) = 5^-1/5 = ?

I'll let you finish that off. All you have to remember is that 5^(-2) 
is really just 5^(-1) divided by 5.

Hope this helps. If you have other questions about this or you're 
still stuck, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/