Square Roots and Fractions

Date: 10/25/98 at 17:57:56
From: Billy Dickson
Subject: Dividing Equations

How do you simplify a square root with a fraction? I have problems 
like:
        ___
       / 1
   _  / ---
    \/   2

Help!
Billy Dickson

Date: 10/27/98 at 08:49:33
From: Doctor Peterson
Subject: Re: Dividing Equations

Okay, Billy, presumably what you are supposed to do is to simplify the 
expression. There's nothing you can do to eliminate the square root 
(other than actually evaluating it, which would be only an 
approximation), but we can simplify it so that we only take the square 
root of an integer (not a fraction), and all square roots are in the 
numerator (so that when we evaluate it we won't have to divide by a 
lot of digits). Simplifying radical expressions like this sometimes 
also requires making the number inside the square root sign as small 
as possible, but we won't have to do that in this problem.

In e-mail, we write the square root as "sqrt", so your problem becomes:

   Simplify sqrt(1/2)

What we do is to first use the rule that: 

   sqrt(a/b) = sqrt(a) / sqrt(b)

so we have:

               sqrt(1)      1
   sqrt(1/2) = ------- = -------
               sqrt(2)   sqrt(2)

Now we only have the square root of an integer, but it's in the 
denominator. How can we move it? If we multiply sqrt(2) by itself, we 
get an integer, 2; so let's multiply both numerator and denominator by 
sqrt(2), which won't change the value:

      1          1 * sqrt(2)     sqrt(2)
   ------- = ----------------- = -------
   sqrt(2)   sqrt(2) * sqrt(2)      2

That's the answer. If you want to go on and evaluate it, you don't even 
need a calculator, if you know that approximately sqrt(2) = 1.414. This 
tells us that sqrt(1/2) is about 0.707. That would have been much 
harder to figure out without simplifying first.

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