Exponents in Algebra

Date: 8/8/96 at 9:6:45
From: Ralph Ware
Subject: Re: Algebra Terms

Dear Dr. Math:

I continue to study for my upcoming placement test and I really 
appreciate all the help and advice that the folks at the Dr. Math 
forum have given me.  Now I am studying the properties of exponents 
from a book about college algebra and doing the exercises.

One of the exercise questions is, "Evaluate: (2^-3)-(2^-2)."  
Well,  the answer is, obviously, -1/8 because (2^-3) is 1/8 and 
(2^-2) is 1/4 and 1/8-1/4= -1/8.  That's all well and good when 
dealing with a simple base like 2 and a simple exponent like -3, 
but I don't understand the process for arriving at the answer.

For example what if the exercise question was, instead, 
"Evaluate:(x^-16)-(x^-18)," or the question involved two dissimilar 
bases (like x and y) or exponents of unknown value represented by 
letters?  Surely there is a procedure for evaluating all such 
expressions.  Could you show me?

Thank you,
   Ralph Ware

Date: 8/25/96 at 23:46:47
From: Doctor Mike
Subject: Re: Algebra Terms

Hello,

What we need to get into here is Common Denominators.  Probably there 
is an index entry on this in your book.  These problems you have 
listed seem to be testing your understanding of negative exponents, 
which seems to be good, but an understanding of this other area is 
holding you back.
   
When you subtract 1/8 and 1/4 you should think of this as 1/8 and
2/8.  Whenever the denominators are the same you just add/subtract
the numerators. The 8 in this case is the common denominator since 
it is common to both fractions (after one of them was re-written).
In general subtract a/x - b/x to get (a-b)/x.  Often what you must
do to get 2 fractions to have the same denominator is to multiply
one or both of them by one, not just any old one, but by a special
form of one.  Here is what I mean for your first case : 

        1     1       1      1     2        1     2         1
       --- - ---  =  --- - (--- * ---)  =  --- - ---  =  - ---
        8     4       8      4     2        8     8         8 
  
The particular value of one used here is 2/2.  In your other one
the 2 denominators are x^16 and x^18.  The common denominator
here is x^18.  So how can you start with 1/(x^16) , multiply it
by one, and get something with x^18 as a denominator.  Answer is
to choose (x^2)/(x^2) as the form of one.  Then 1/(x^16) times 
(x^2)/(x^2) becomes (x^2)/(x^18).  Keep in mind that this new
fraction is THE SAME AS 1/(x^16) since all we did was to multiply
by one.  Now your original (x^-16)-(x^-18) has been re-written
with a common denominator as (x^2)/(x^18) - 1/(x^18) so the 
difference is (x^2-1)/(x^18).  I cannot give you a total lesson
in this area now, but this should get you started. Write back if
you encounter another sticking point.  I hope this helps.  
  
-Doctor Mike,  The Math Forum
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