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### 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

more, or if you have any other questions.

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

Associated Topics:
Middle School Exponents
Middle School Factoring Numbers

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