Cancelling Exponents

Date: 03/04/97 at 19:03:24
From: Philip Powell
Subject: Multiplying nasty fractions

Hello,

How did they turn   2^29 x 3^16          3
                    ------------  into  --- ?
                    2^32 x 3^15         2^3
                    

Any help would be greatly appreciated. I do have one other question 
though: How do you do this for free? Our family is quite amazed that 
you have the time to answer all of these questions and that you do it 
for no charge. (That's not a complaint, mind you!)

Date: 03/04/97 at 23:40:21
From: Doctor Mike
Subject: Re: Multiplying nasty fractions

Hello,     

Think about this formula:
  
        A     X       A*X 
       --- * ---  =  -----
        B     Y       B*Y 
  
In a way that's really all you need, together with the definition of 
exponent, and the fact that  N/N = 1.  
  
    2^29 x 3^16     2^29   3^16     2^29    1    3^15    3 
    -----------  =  ---- x ----  =  ---- x --- x ---- x ---
    2^32 x 3^15     2^32   3^15     2^29   2^3   3^15    1 
  
Now anything where the numerator equals the denominator just 
disappears because it is the same as multiplying by one, so that long 
expression to the right above equals:

         1         3       1     3       3       3  
    1 x --- x 1 x ---  =  --- x ---  =  ---  =  --- 
        2^3        1      2^3    1      2^3      8 
   
The safest way to work complicated problems like this is to take lots 
of "small steps" which you know are correct because of some really 
basic principles.  I should tell you that there are some "shortcuts" 
or "tricks" to doing this kind of problem faster.  You might hear 
someone say something like "JUST CANCEL THE 2^29 INTO THE 2^32 LEAVING
EIGHT IN THE DENOMINATOR".  There is nothing wrong with that if you 
really know what you are doing.  But if you are a little bit unsure or 
rusty, you can get into trouble that way.
  
I don't answer Dr. Math questions for a living.  It's sort of a hobby.  
Besides, I think it's good to volunteer your efforts as you can.  I am 
not the only mathematician on staff here.  There's quite a bunch of 
us, all with different interests.  Even so, we don't get to all the
questions.  That's too bad because some question we don't get to might 
be from a student who is really struggling, or from a young genius 
needing to be taken seriously and given encouragement.  
  
Anyway, enough philosophy.   I hope you got what you need. 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/