Subtracting Exponents when Dividing Like Bases

Date: 01/21/2005 at 16:36:27
From: snarles
Subject: my question is about simplifying variables

I need to simplify x to the seventh power over x to the third power.  
I think I should divide 7 by 3 and put that as the power, but I end up 
with x to the 2.3 repeating power, and the book says it's x to the 
fourth power.

Date: 01/21/2005 at 18:10:11
From: Doctor Achilles
Subject: Re: my question is about simplifying variables

Hi Snarles,

Thanks for writing to Dr. Math.

So your problem is:

  x to the seventh
 ------------------
   x to the third

Now, remember that x to the seventh is shorthand for x*x*x*x*x*x*x

(The * is a multiplication symbol.)

So your problem is really:

  x*x*x*x*x*x*x
 ---------------
      x*x*x

Ok, so the first step in reducing a fraction is to find a common 
factor in the top and bottom.  One common factor is x:

  x*(x*x*x*x*x*x)
 -----------------
  x*(x*x)

Now we cancel out the common factor and we are left with:

  x*x*x*x*x*x
 -------------
      x*x

Now find another common factor:

  x*(x*x*x*x*x)
 ---------------
  x*(x)

And we get:

  x*x*x*x*x
 -----------
      x

And again, we find a common factor:

  x*(x*x*x*x)
 -------------
  x*(1)

And we get:

  x*x*x*x
 ---------
     1

Or just:

  x*x*x*x

Or just:

  x to the fourth power

So that is the answer.

You may have heard the rule "When you divide like bases with 
exponents, subtract the powers" or something like that.  What that
means is when you have something like:

  x to the seventh
 ------------------
   x to the third

Your answer will be

  x to the (7-3)

Or:

  x to the fourth

Hope this helps.  If you have other questions or you'd like to talk
about this some more, please write back.

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