Rationalizing Denominators with Multiple Radicals

Date: 11/04/2004 at 04:09:57
From: chi thien
Subject: i need help

How can I rationalize the denominator of a fraction when it contains
many square roots, such as:

                      1
-------------------------------------------------
sqrt(3) + sqrt(5) + sqrt(7) + sqrt(11) + sqrt(13)

Date: 11/04/2004 at 16:47:59
From: Doctor Vogler
Subject: Re: i need help

Hi Chi,

Thanks for writing to Dr. Math.

You have to do this in five steps, and the coefficients will get 
pretty big.  But that's the way it happens.  At each step, you pick a
prime (start with p = 3, then use p = 5, and then 7, 11, 13) and the
idea is that you write your fraction in the form

  q / (r + s*sqrt(p))

where r and s only have square roots without any sqrt(p)'s in them. 
Then you multiply the numerator and denominator by r - s*sqrt(p) in
order to change this to:

  q*(r - s*sqrt(p))
  ----------------
    r^2 - s^2*p

and you've gotten rid of the sqrt(p)'s in the denominator!  It might
help you get things in the right place if you left products of square
roots in the form

  sqrt(3)*sqrt(7)

(at least in the denominator) since you'll just have to factor them
out later anyway.  But multiply together double square roots, as in

  sqrt(3)*sqrt(3)  becomes 3.

When the denominator is finally an integer, then you're done.

So, the first time you factor, you get

  (sqrt(5) + sqrt(7) + sqrt(11) + sqrt(13)) + sqrt(3)*(1)

and then the second time, you have

  33 + 2sqrt(5)sqrt(7) + 2sqrt(5)sqrt(11) + 2sqrt(5)sqrt(13) +
        2sqrt(7)sqrt(11) + 2sqrt(7)sqrt(13) + 2sqrt(11)sqrt(13)

which you then factor into

  (33 + 2sqrt(7)sqrt(11) + 2sqrt(7)sqrt(13) + 2sqrt(11)sqrt(13))
           + sqrt(5)*(2sqrt(7) + 2sqrt(11) + 2sqrt(13))

And so forth.  Can you finish it up?

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

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

Date: 11/06/2004 at 07:33:39
From: chi thien
Subject: Thank you (i need help)

Thank you!  You helped me so much!