Five to the Third Power

Date: 09/20/2001 at 17:17:03
From: Daniel
Subject: Powers/exponents

Dear Dr. Math,

The one thing that is giving me most trouble is powers and exponents, 
such as:

5 to the 3rd power is? On a test I put 15. Wrong!

Help,
Daniel

Date: 09/21/2001 at 14:30:53
From: Doctor Ian
Subject: Re: Powers/exponents

Hi Daniel,

Have you learned about prime factoring yet? For example, 36 has lots 
of factors:

  72 = 1 * 72
  36 = 2 * 36
     = 3 * 24
     = 4 * 18
     = 6 * 12
     = 8 *  9

But it can be reduced to a set of prime factors:

  72 = 2 * 2 * 2 * 3 * 3

and these factors can be used to produce all the other factors:
 
  2 * 2 = 4
  2 * 3 = 6
  2 * 2 * 2 = 8
  2 * 2 * 3 = 12
  2 * 2 * 3 = 18
  2 * 2 * 2 * 3 = 24
  2 * 2 * 3 * 3 = 36

Now, it's not very convenient to write

  72 = 2 * 2 * 2 * 3 * 3

so it's somewhat natural to look around for a more compact notation.  
One such notation would be to count the number of times that a prime 
factor appears, and use that number instead of repeating the factor 
that many times. Using this notation, 

        3   2
  72 = 2 * 3

because 2 appears three times, and 3 appears twice.  

Let's look at how another factorization might be written:

                                            2    4
  2 * 3 * 4 * 4 * 5 * 5 * 5 * 5  = 2 * 3 * 4  * 5

Note that when something appears only once, it gets no exponent, 
although we _could_ use the exponent 1:

                                    1    1    2    4
                                 = 2  * 3  * 4  * 5

But that makes more work, not less. Note also that when you get larger 
and larger collections of the same factor, this can really save a lot 
of writing. For example, would you rather write

  2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

or

   13
  2   ?


It looks as though, during your test, you saw the 5 and the 3 together 
and thought you were supposed to multiply. And that's a very natural 
thing to think, if you aren't familiar with the notation for 
exponents. But where

  5 * 4

would be read '5 times 4', and indicates that you're supposed to 
multiply 5 by 4, 

   4
  5

would be read '5 to the 4th power', and indicates that you're supposed 
to multiply 5 by 5 four times in a row. 

Once you've got the notation down, the next step is to get a handle on 
the basic properties of exponents. You can find a gentle introduction 
to them here:

  Properties of Exponents
  http://mathforum.org/dr.math/problems/crystal2.01.22.01.html   

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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