I'm with Guy. It's a convention, and one that is hard to remember (for students).
I know we want it read as the opposite of 3 squared, and we can look at -3 as -1 * 3 if we want, though that depends on your base axioms, and usually the negatives come from positing an axiom that every number has an additive inverse, and someone else said we can look at it as subtraction, 0 - 3^2. But to me these are all patches, that manifestly don't work well. None of the suggestions I see for finding a rational basis for one interpretation are widely accepted conventions, it seems to me.
I'm mindful that a $250M spacecraft crashed into Mars instead of orbiting because in the US we used feet and inches, while in England they used metric. I wouldn't want to fly in a plane designed where one engineer used the expression -3^2 (I know this is a stretch) and sent it to the factory for fabrication. Very iffy. As Guy notes we should write what we mean clearly. It is a convention that should be avoided, perhaps banned. :) I don't waste time on it and don't test it.
So, -x^2 evaluated for x = -3, fine. But not -3^2. And even the expression with the variable could be dangerous.
Actually the only good rule I see is that an exponent only applies to the symbol it "touches". Not very mathematical but it seems to work.
p.s. I guess my being peevish about it is that it takes much more class time than it is worth (usually starts a polite argument with the students), and if you want to ensure few perfect papers, do put it on the test. OK, enough from me on the subject. :)
________________________________ From: Wayne Ford Mackey [mailto:email@example.com] Sent: Thursday, October 18, 2012 12:11 PM To: Guy Brandenburg; firstname.lastname@example.org; Phil Mahler Cc: mathedcc Subject: RE: Please remind me why -3^2 = -9
It should be read as the opposite of 3 squared. Since 3 squared is 9, the opposite is -9. The "-" sign is used in 3 different ways. In front of a natural number it means negative or minus, in front of anything else it means opposite and between two things it means add the opposite.
________________________________ From: email@example.com [firstname.lastname@example.org] on behalf of Guy Brandenburg [email@example.com] Sent: Thursday, October 18, 2012 6:04 AM To: firstname.lastname@example.org; Philip Mahler Cc: mathedcc Subject: Re: Please remind me why -3^2 = -9 It's a convention. In a case like that, one really ought to use parentheses to make the meaning clear, since a lot of people, not just youngsters, will get confused.
If one intends to say (-3)*(-3), then write (-3)^2. If one means - (3)*(3), then write - (3^2).
-3 means -1 x 3, so -3^2 is (-1)(3^2) = (-1)(9) = -9. John Peterson
________________________________ From: "Philip Mahler" <email@example.com> To: "mathedcc" <firstname.lastname@example.org> Sent: Thursday, October 18, 2012 5:37:07 AM Subject: Please remind me why -3^2 = -9
I have been teaching a long time, and I know from experience that 50% of students will tell me that -3^2 = +9 on a test or a final, despite having discussed it a few times in a course.
When I first started teaching I taught calculus and precalc. Piece of cake. Then I started with an Algebra I class and couldn't connect at all for the first week or so. I was ready to believe I couldn't teach. I simply could not explain how I got the right answers when evaluating expressions... Then I discovered the order of operations (PEMDAS to some). A definition of the order of operations which I had so internalized that I didn't know there was a rule for it. Since that discovery I've been a wonderful teacher. :-)
So... I must be missing something that so many of my students think -3^2 is +9. What is the rule I have never discovered?
Full disclosure: I think -k^2, k a constant, should be banned from mathematics texts and tests. -x^2, x a variable, evaluated for say -3, absolutely (no pun intended) but not -3^2.