Dividing with Powers

Date: 01/16/2003 at 21:17:20
From: John Doe
Subject: Dividing 
                                                    7
I am having trouble with problems like this 6.0 x 10
                                           ----------
                                                    2
                                            3.0 x 10
Will you please help?!?

I just don't know what to multiply or divide or add or subtract from 
what.

Date: 01/17/2003 at 13:28:58
From: Doctor Ian
Subject: Re: Dividing 

Hi John,

Let's consider a slightly different problem. Suppose we wanted to 
divide

  6 * 10
  ------
  3 *  5

We could multiply everything out first to get

  6 * 10   60
  ------ = -- = 4
  3 * 5    15

But we could also do this:

  6 * 10   6   10
  ------ = - * -- = 2 * 2 = 4
  3 *  5   3    5

That is, the rule for multiplying fractions is 

  a   b   a * b
  - * - = -----
  c   d   c * d

but we can use the same rule to combine fractions, or to break them up
into separate fractions.  If this doesn't make sense to you, please
let me know and I'll try to find another way to explain it, because
it's a very important idea. 

Now, let's look at your example, and use the same idea:

  6.0 * 10^7   6.0   10^7
  ---------- = --- * ----
  3.0 * 10^2   3.0   10^2
 

                   10^7
             = 2 * ----
                   10^2

So, what do we do with the powers of 10?  There are a few ways to
think about it.  One is to just use the definition of exponents:

  10^7   10 * 10 * 10 * 10 * 10 * 10 * 10
  ---- = --------------------------------
  10^2   10 * 10

         10 * 10   
       = -------- * (10 * 10 * 10 * 10 * 10)
         10 * 10            

       = 1 * (10 * 10 * 10 * 10 * 10)

       = 10^5

All we did, really, was cancel out some of the 10's in the exponent. 
A quicker way to do that is by crossing out zeros:

               xx
  10^7   10000000
  ---- = -------- = 100000 = 10^5
  10^2        100
               xx

(Note that with powers of 10, the exponent is the same as the number
of zeros following the 1.)

Do you see why it's really the same thing?  There's an even quicker
way to do it, which is to subtract the bottom exponent from the top 
one:

  10^7   
  ---- = 10^(7-2) = 10^5
  10^2   

After you've been doing this a while, that's probably the method 
you'll eventually decide to use. But there's no need to rush. 

Anyway, the main idea is that you deal with the powers of 10
separately:         

  6.0 * 10^7   6.0   10^7
  ---------- = --- * ---- = 2.0 * 10^5
  3.0 * 10^2   3.0   10^2

This, by the way, is one of the reasons we use powers of 10. It's much 
easier to divide one small number by another, and deal with the
exponents using subtraction, than it is to divide one big number by
another big number. 

Does that make sense? 

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