Decimal Exponents

Date: 11/07/2001 at 17:20:09
From: Alex
Subject: Exponents

Dear Dr. Math,

I was sitting in math class today and I asked my teacher a question. 
She was not sure and told me to come to your Web site. Here's the 
question: 

Can you have exponents that are decimals? 

My teacher said that maybe they have it in the high math classes. Do 
you know?

Respectfully,
Alex, 6th grade

Date: 11/08/2001 at 14:45:57
From: Doctor Peterson
Subject: Re: Exponents

Hi, Alex.

Good question! Your teacher is right; as you advance in math you will 
see more advanced kinds of exponents.

When you first see exponents, only whole numbers are used. Then we 
introduce negative and zero exponents:

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

   Zero as an Exponent - Dr. Math archives
   http://mathforum.org/dr.math/problems/gerleman2.1.96.html   

Briefly, for example, 2^-3 (that is, 2 to the -3 power) means the 
reciprocal of 2^3, which is 1/8.

After that, we can talk about rational (fraction or decimal) 
exponents:

   Finding a Decimal Exponent
   http://mathforum.org/dr.math/problems/tyler3.21.98.html   

   Resolving Decimal Exponents
   http://mathforum.org/dr.math/problems/adel.3.26.01.html   

Again briefly, 2^0.5, or 2^(1/2), the one-half power of 2, is the same 
as the square root of 2. You can see this by thinking about what the 
square of 2^(1/2) would be.

Finally, we can talk about irrational exponents, like 2 raised to the 
pi power (and if you've ever heard of "imaginary numbers," we can use 
them, too):

   Meaning of Irrational Exponents
   http://mathforum.org/dr.math/problems/knobler.three.3.17.97.html   

This can't be explained briefly!

Each step requires us to do a little new thinking, so it takes time to 
develop an understanding of all this.

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