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### Rationalizing the Denominator

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Date: 07/20/2001 at 00:26:25
From: Matt Sellers
Subject: Rules for fractions w/ square roots

My teacher and I are having a discussion about square roots in the
denominator. I have always been told that you must take out any square
roots in the denominator. Over time I have accepted it as a rule of
math. Am I wrong to believe that for a final answer using fractions
you should take out the square roots from the denominator? If I am
right, where can I find the rule so I may show my teacher?

Thanks,
Matt
```

```
Date: 07/20/2001 at 12:32:52
From: Doctor Rob
Subject: Re: Rules for fractions w/ square roots

Thanks for writing to Ask Dr. Math, Matt.

"Rationalizing the denominator" has been taught in schools for many,
many years. See any older algebra book. This dates back to the time
when computations had to be done by hand. In this situation, 1/sqrt(2)
is harder to compute in decimal form than its equal, sqrt(2)/2. Both
require the extraction of the square root, but the first involves a
much more difficult division operation than the second. Thus it was
deemed that the second form was "simpler" than the first. When
simplifying, students were always taught to reduce the expression to
its simplest form, which invariably included rationalizing any
denominators, wherever possible.

In the modern world, where such calculations are usually done by
computers or calculators, it is not clear that such "simplification"
makes sense any more.

A more complicated example of this can easily occur. For example,
if cbrt(x) means the cube root of x, I think that

1/(1+cbrt[2])

is simpler than its equal

(1-cbrt[2]+cbrt[4])/3.

On the other hand, it is clear that

1 + cbrt(2)

is simpler than its equal

3/(1-cbrt[2]+cbrt[4]).

Sometimes simpler expressions result from rationalizing, and
sometimes from not rationalizing.

Furthermore, if you have nested radicals, things get even worse.  I
don't even want to write down the rationalized-denominator form of

1/(1+sqrt[3-sqrt(1+sqrt[7]))]).

On the other hand, rationalizing denominators does tend to produce one
single, well-defined "simplest" form. When you do this, you can easily
compare two expressions in simplest form to see whether or not they
are equal.

So, you see, there are arguments on both sides of this question.

I hope that this satisfies your need.  If not, write again.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents

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