Exponent of Negative Integer

Date: 07/27/97 at 16:02:14
From: James Middleton
Subject: Exponent of Negative Integer

Hi, I'm a grade 8 teacher in Ontario, Canada and we are having a 
debate among staff and students about what the value of -5^2 
(neg five squared) is.

In our grade 8 teaching, we have always taught this in line with the 
usual rules of signs, so for us -5^2 = -5 * -5 = +25.  This is also 
how it is shown in our math textbooks.

However, students who go on to a local high school for grade 9 keep 
coming back to us saying the H.S. math teachers say -5^2 = -25.

Who is right, and why?

I understood the definition of a polynomial in the form x^n to be that 
if x is a real number and n is a natural number, then

       x^n = x1 * x2 * x3 ... * xn

In this case if x = -5 then x^2 = -5 *-5 = +25.

But the H.S. math teachers, say -5 is really (-1)(5) so to them
-5^2 = (-1) * (5^2) = (-1) * 25 = -25. We just don't see it.

They say there is a problem in evaluating an expression like: 32-5^2  
Does this mean 

     32 - (5^2) = 32 - (25) = 7, 

or does it mean 

     32 + (-5^2) = 32 + (-25) =7?  

To us they are the same.

It would be another case if the question was -(5^2), which we assume 
would have a value of -25.  What about -(5)^2?

Anyway, all this is driving us all nuts. Your help would be 
appreciated. Thanks.

James Middleton

Date: 07/28/97 at 16:41:48
From: Doctor Anthony
Subject: Re: Exponent of Negative Integer

The correct answer depends on asking the correct question.  

If you ask 'What is the square of -5 ?'  the answer is  +25.
In symbols this is (-5)^2 = +25

If you ask 'What is - 5^2 ?' the answer is -25 
In symbols this is  - (5)^2 = -25

It is a matter of being careful how you write and word the question. 
It should never be so expressed that there is doubt as to which 
meaning applies.  If there is doubt, the question should be reworded.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/