Exponents

Date: 12/04/96 at 09:12:48
From: Michael Baker
Subject: Exponents

Dear Doctor Math,

I have trouble working out exponent problems.  Is there an easier way 
to work them or understand them?

Date: 12/04/96 at 13:30:01
From: Doctor Mike
Subject: Re: Exponents

Hi Michael,
  
It's great to hear from another "Mike".  I'm glad you asked this 
because there IS a good way to figure out exponent problems.  It is 
really such a simple idea that it's really easy to forget to do.  

ALWAYS KEEP THINKING ABOUT EXACTLY WHAT THE EXPONENT MEANS.  For 
instance, say you are asked to simplify 9^3 times 3^9 which is usually 
written in books as : 
                                       3    9    
                                      9  * 3  
      
but we usually write it in E-mails as 9^3*3^9 to save space.  
  
Let's start by asking what 9^3 means.  It is 3 nines multiplied 
together, or 9*9*9. I'm sure you can figure out what that is, but we 
don't need to know that. In fact we need to go in the OTHER direction 
and break it up MORE. Since 9=3*3, we know 9*9*9 is really 
(3*3)*(3*3)*(3*3), which is 3*3*3*3*3*3.  

Now it's time to think again about the meaning of exponents to see 
that this is exactly the same as 3^6 or 3 to the sixth power.  Right?  
Now we can re-write the original problem as : 

                                       6    9 
                                      3  * 3 
   
If the goal of the problem is to simplify, then I think we have made 
progress, since we are now dealing with only one number raised to some 
exponents. But we can do more. Time again to think about what the 
exponent means. The left side is 6 three's multiplied together, and 
the right side is 9 three's multiplied together. We can actually 
write that out to see what it looks like: 
  
         (3*3*3*3*3*3)*(3*3*3*3*3*3*3*3*3)
  
How many three's are there?  Go ahead and count them. There are 15
of them. So how can you write down that you have 15 three's all
multiplied together?  Yes, 3 to the 15th power or 3^15.  Here I have
once again just used the same old information about what the exponent
notation MEANS.  So, we now know that 9^3*3^9=3^15.   
  
Eventually you will start doing a lot of this right in your head, but 
you should always keep reminding yourself what it all means.  This 
will be especially important for problems a lot harder than the one we 
just did.  
      
I hope this helps.  Bye for now.   

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