Why Simplify Square Roots?

Date: 05/17/99 at 10:38:09
From: John Cendrowski
Subject: Simplifying square roots--WHY?

In a discussion with colleagues, the question arose regarding the 
rationale behind teaching simplification of radicals.  We suggested 
addition, simplifying roots, and working with complex numbers.

My colleague is still not convinced. He says students think of root 
32 as a number between 5 and 6, not as 4 root 2.  

All three of us have been trained to simplify square roots, but we are 
now looking at the reasons why we should continue to teach the 
process.

Thank you.

Date: 05/17/99 at 13:28:12
From: Doctor Rick
Subject: Re: Simplifying square roots--WHY?

Hi, John. Thanks for the question.

For some purposes - when all you want is a numerical answer to a 
single question - it may be sufficient to punch 32 into the calculator 
and see that it's 5.656854249. For some purposes, the most important 
thing about this number is that it lies between 5 and 6. But the real 
power of math - the power for understanding - is in its exactness. 

If you look at math history, you will find that the real innovation in 
Greek mathematics, compared with earlier Babylonian and Egyptian math, 
was that they made a distinction between approximations and exact 
results.

The fact that two numbers are approximately equal, even to 9 decimal 
places, doesn't have any profound meaning. The number 5.656854249 
doesn't mean anything to me, and in fact it ISN'T anything particular 
- just an APPROXIMATION to the square root of 32, to an arbitrary 9 
decimal places. 

On the other hand, 4*sqrt(2) is an EXACT answer, with infinite 
precision; one can tell whether another number is EXACTLY equal to it. 
The expression can bring specific mathematical associations to my mind 
- for instance, it could be the diagonal of a square of side 4. Also, 
if the answer is not correct, I can tell a lot more about the nature 
of the student's error from the radical form than I could from a 
decimal.

Thus, there is real mathematical benefit in retaining the exact value 
of an expression. For irrationals, this is only possible if a radical 
is retained in the expression - any finite decimal, which is 
necessarily a rational number, will be only an approximation.

Simplification of a radical has the same value as reduction of a 
fraction to lowest terms. It permits us to recognize that two radical 
expressions are equivalent, even if their original forms appeared 
quite different. In particular, it allows a teacher to tell at a 
glance whether the answer is correct.

I hope this gives you a rationale for passing on your training to the 
next generation. In the computer generation, much can be done with 
numerical methods; but real UNDERSTANDING of a problem calls for exact 
answers. Exact answers call for retension of radicals (along with pi 
and e), and recognition of a particular radical expression calls for 
simplification.

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