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

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