Simplifying Fractions with Radicals

Date: 03/09/2006 at 21:38:19
From: Sandy
Subject: simplifying square roots

How do I simplify 18 sqrt 200 divided by 12 sqrt 500?

Date: 03/09/2006 at 23:20:06
From: Doctor Ian
Subject: Re: simplifying square roots

Hi Sandy,

Do you mean
       ____
  18 \/ 200
  --------- ?
       ____
  12 \/ 500

Let's start by noting that because of the way we multiply fractions,
we can write this as

          ____
  18    \/ 200
  -- * -------- 
          ____
  12    \/ 500

Do you see why?  So if we can simplify each of these fractions
separately, we'll be in good shape.  Okay?  

For the first one, we want to find the prime factors of the numerator
and denominator:

  18   2 * 3 * 3
  -- = ---------
  12   2 * 2 * 3

How does this help us?  Again, breaking things into products of
fractions (do you see how we just keep using the same tricks over and
over again?), we get

  18   2 * 3 * 3   
  -- = ---------
  12   2 * 2 * 3   

       2   3   3
     = - * - * -
       2   2   3

           3   
     = 1 * - * 1
           2   

       3   
     = -
       2   

So that's how we can simplify the first part.  Does that make sense? 

So, what about that second part?  Well, we'll do basically the same
thing, but inside the radicals:

     ____      __________________
   \/ 200    \/ 2 * 2 * 2 * 5 * 5
  -------- = --------------------
     ____      __________________
   \/ 500    \/ 2 * 2 * 5 * 5 * 5

How does this help?  Well, an important thing to know about square
roots is that 
     ______     __     __
   \/ a * b = \/ a * \/ b

So we can use our little trick again, to get

        __________________
      \/ 2 * 2 * 2 * 5 * 5
   = ----------------------
        __________________
      \/ 2 * 2 * 5 * 5 * 5

        __     __     __     __     __
      \/ 2   \/ 2   \/ 2   \/ 5   \/ 5
   =  ---- * ---- * ---- * ---- * -----
        __     __     __     __     __
      \/ 2   \/ 2   \/ 5   \/ 5   \/ 5

                __   
              \/ 2   
   =  1 * 1 * ---- * 1 * 1 
                __     
              \/ 5  

        __   
      \/ 2   
   =  ----  
        __     
      \/ 5 

Put it together, and we get

         ____           __
  18    \/ 200    3   \/ 2
  -- * -------- = - * ---- 
          ____    2     __
  12    \/ 500        \/ 5

                      __
                  3 \/ 2
                = ------
                      __
                  2 \/ 5

Were you able to follow all that?

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

Date: 03/10/2006 at 03:34:41
From: Sandy
Subject: Thank you (simplifying square roots)

Thank you sooo sooo soooooo much!  You have made my day and now I get
how to do all my questions.  Keep up the good work!