Rationalizing the Denominator

Date: 07/20/2001 at 00:26:25
From: Matt Sellers
Subject: Rules for fractions w/ square roots

My teacher and I are having a discussion about square roots in the 
denominator. I have always been told that you must take out any square 
roots in the denominator. Over time I have accepted it as a rule of 
math. Am I wrong to believe that for a final answer using fractions 
you should take out the square roots from the denominator? If I am 
right, where can I find the rule so I may show my teacher?
  
Thanks, 
Matt

Date: 07/20/2001 at 12:32:52
From: Doctor Rob
Subject: Re: Rules for fractions w/ square roots

Thanks for writing to Ask Dr. Math, Matt.

"Rationalizing the denominator" has been taught in schools for many,
many years. See any older algebra book. This dates back to the time 
when computations had to be done by hand. In this situation, 1/sqrt(2) 
is harder to compute in decimal form than its equal, sqrt(2)/2. Both 
require the extraction of the square root, but the first involves a 
much more difficult division operation than the second. Thus it was 
deemed that the second form was "simpler" than the first. When 
simplifying, students were always taught to reduce the expression to 
its simplest form, which invariably included rationalizing any 
denominators, wherever possible.

In the modern world, where such calculations are usually done by
computers or calculators, it is not clear that such "simplification"
makes sense any more.

A more complicated example of this can easily occur. For example,
if cbrt(x) means the cube root of x, I think that

   1/(1+cbrt[2])

is simpler than its equal

   (1-cbrt[2]+cbrt[4])/3.

On the other hand, it is clear that

   1 + cbrt(2)

is simpler than its equal

   3/(1-cbrt[2]+cbrt[4]).

Sometimes simpler expressions result from rationalizing, and
sometimes from not rationalizing.

Furthermore, if you have nested radicals, things get even worse.  I
don't even want to write down the rationalized-denominator form of

   1/(1+sqrt[3-sqrt(1+sqrt[7]))]).

On the other hand, rationalizing denominators does tend to produce one 
single, well-defined "simplest" form. When you do this, you can easily 
compare two expressions in simplest form to see whether or not they 
are equal.

So, you see, there are arguments on both sides of this question.

I hope that this satisfies your need.  If not, write again.

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