Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Rationalizing Denominators


Date: 07/12/97 at 15:11:49
From: Jim Summerlin
Subject: Rationalizing denominators

Dear Dr. Math,

HELP!!  I just can't seen to do this.  How can I rationalize a 
denominator with odd power radicals?  This one for example

   1/[ (64^(1/5))+(36^(1/5))+(24^(1/5))+(16^(1/5))]

It looks better with radicals but I don't know how to enter symbols in 
this text box.  Can you rationalize that denominator?  Thanks for your 
help.

Jim Summerlin


Date: 07/16/97 at 16:19:06
From: Doctor Rob
Subject: Re: Rationalizing denominators

Yes, you can, but it will end up with a ghastly numerator.  

In order to do this, you need to drag in a 5-th root of unity, call it 
z. Then it satisfies (z^5 - 1)/(z - 1) = z^4 + z^3 + z^2 + z + 1 = 0.  
You then multiply both numerator and denominator by the product of all

  z^a*(64^(1/5))+z^b*(36^(1/5))+z^c*(24^(1/5))+z^d*(16^(1/5))

for all 4-tuples (a, b, c, d) with 0 <= a, b, c, d <= 4, except for 
the 4-tuple (0,0,0,0), which is already there in the denominator.  
This is a list of 255 additional factors.  Now expand everything in 
sight, and simplify using first z^5 = 1 and then the above irreducible 
quartic polynomial equation satisfied by z.  

It may be hard to believe, but all the z's will miraculously vanish 
from the expression, and all the radicals will vanish from the 
denominator, and you will have your answer. Of course not all the 
radicals vanish from the numerator!

The denominator turns out to be

    2551318400000^25,

or about 5.831779 * 10^303 !

In this particular case, we didn't need b or d at all, because

  (64^(1/5))^4 = 16*16^(1/5)

and

  (24^(1/5))^2/(16^(1/5)) = 36^(1/5).

If we hadn't used the b and d, we would have gotten a denominator of
only 2551318400000. After reducing to lowest terms, the denominator
is only 510263680.

Computing the numerator I leave as an exercise ( :-) !).  Actually
it only consumed 7 lines of Mathematica output.

Convinced?

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
http://mathforum.org/dr.math/
Associated Topics:
High School Basic Algebra

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/