Factor and Simplify

Date: 4/10/96 at 21:41:35
From: Anonymous
Subject: Complex Fractions

How would you do the following problem and if possible, please 
give tips for how to do similar problems.  Thank you for the help.

(1/x^2-1/y^2)/(1/x^2+2/xy+1/y^2)

Factor whenever possible.

Date: 5/28/96 at 20:33:24
From: Doctor Pete
Subject: Re: Complex Fractions

Let a=1/x, b=1/y.  Then your expression is equivalent to

  (a^2-b^2)/(a^2+2ab+b^2)

which you can factor as (a+b)(a-b)/(a+b)^2, so this is 
(a-b)/(a+b).

Substituting back, we get (1/x-1/y)/(1/x+1/y) = 
((y-x)/xy)/((x+y)/xy), so your expression is equal to 
(y-x)/(x+y).

The trick in this case was to look at your x's and y's, and notice 
that they were all reciprocals, so you can do a substitution that 
makes the factorization much easier to spot.  Here's one in a 
similar vein, but a bit trickier:  

Simplify

  (x-y)/(sqrt(x)-sqrt(y)) .

-Doctor Pete,  The Math Forum