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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 

Can you have exponents that are decimals? 

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

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   

   Zero as an Exponent - Dr. Math archives   

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) 

   Finding a Decimal Exponent   

   Resolving Decimal Exponents   

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   

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   
Associated Topics:
High School Exponents
Middle School Exponents

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