Algebra and Exponents: -4^2 = ?

Date: 06/30/98 at 21:48:00
From: David Eisner
Subject: Algebra and exponents

I had an argument with a professor over this topic...this was the 
question on the test:

  -4^2 = ?

I say 16, He says -16. Now, there are no parentheses around the 4 to 
indicate that it is in fact -1 * 42. He has told me the parentheses 
are implied. If I have learned one thing in math it is that nothing is 
implied, but rather it is simply solved. Thank you for your help.

David

Date: 07/01/98 at 08:03:33
From: Doctor Allan
Subject: Re: Algebra and exponents

Hi David,

I will have to agree with your professor on this one. It simply has to 
do with the order of operations. 

Try to remember this one:
 
PEMDAS
       1. Parentheses 
       2. Exponents 
       3. Multiplication and Division 
       4. Addition and Subtraction 

This means that you should do what is possible within parentheses 
first, then exponents, then multiplication and division (from left to 
right), and then addition and subtraction (from left to right).

In your case the question is whether to interpret the expression as 
(-1*4)^2 or (-1)*(4^2).

Since there are no parentheses, you should calculate the exponents 
first before continuing with multiplication (according to PEMDAS). 
That's why the result is -16.

It's a matter of convention. We have to choose one in order to avoid
ambiguity and PEMDAS is the one used. 

I hope this is satisfying to you.

Kind regards,

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

Date: 07/01/98 at 20:45:00
From: David Eisner
Subject: Re: Algebra and exponents

Yes, but that leaves the question open that -1*4 = -1*4, and that -4 
is in all actuality not a number at all. If that does in fact hold 
true, then why not instead of 82 = 64, interpret that as 2*42? Then 
the answer is changed completely. -4 in itself is classified as a real 
number, and can be substituted for X. Also, the answer to the question 
x2 = 16 is {4, -4}, is it not? Substituting either one of your answers 
into the original equation would appear as this: 42 = 16, -42 = 16.  
Now, how can -42 = -16 and 16, when it is clearly proven on the right 
using simple substitution that the answer is 16.

Thank you for your help again.

David

Date: 07/01/98 at 09:19:31
From: Doctor Allan
Subject: Re: Algebra and exponents

>If that does in fact hold true, then why not instead of 82 = 64, 
  interpret that as 2*42? 

You can do that, but when you interpret that way you implicitly place 
the parentheses as (2*4)^2 as your professor pointed out to you.

>Substituting either one of your answers into the original equation
 would appear as this: 42 = 16, -42 = 16.  

And this is precisely the point. When substituting 4 and -4 into 
the original equation you implicitly use parentheses. If you write 
-4^2 you mean (by PEMDAS) -(4^2) no matter what you intend to say. 
Substituting -4 in the expression x^2 means that you should write 
(-4)^2. By using PEMDAS this is so as soon as you choose to 
substitute -4.

Sincerely,

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