This is not homework, but rather what I believe is a mistake in a homeschooling book. If someone asked you to solve (-3^2), and that is exactly how the problem is written with the square inside the parenthesis, wouldn’t you say that is 9? Isn’t that simply the number -3 squared??? They claim the answer is -9.
I disagree with the answer arguing that -3 is indeed a number, and the square of -3 is 9. Must it be written as (-3)^2 for my answer to be correct???
According to the Wiki article, and this jibes with what I remember, the book is right and you are wrong, and exponentiation takes precedent over unary negation. It contains the very example you are using:
Here are some examples to play with…
the order of operations goes - parenthesis, exponents, then multiply/divide, then add/subtract.
a common memnotic is “please excuse my dear aunt sally.” so, yes. you would have to parenthesize the -3 before squaring to get positive 9.
That would be simple enough had the problem been (-3)^2, but with the exponent inside the parnethesis, one cannot truly follow the “please excuse…” rule since one MUST execute the exponent FIRST in order to evaluate what is inside the parenthesis!
You just answered your own question. The exponent is evaluated first, then the negation. Remember that negation is a unary operator (when it comes to order-of-operations problems).
If it helps, consider that (-3[sup]2[/sup]) is just shorthand for (-1 x 3[sup]2[/sup]).
If the book is right, then there IS no such thing as -3, or any negative values, on a numberline. Or, at best, the numberline is only trying to show relative position, but not unique numbers. In other words, the 3 to the left of the zero is not unique from the 3 to the right of zero. Yet, math will contradict itself by claiming that -3 < 3. They can’t have it both ways!!!
Also, if math rules claim it is proper to parse the negative from the three (in my exmaple), that is like saying Jonathan is a really guy named Jo Nathan. That’s simply BS. I fail to see why the “integrity” of -3 is not maintained as a distinct number unto itself.
You pretty much just answered yourself.
Sure, there are two different numbers, 3 and (-3). But in -3^2, what you see is shorthand for “take the number 3 and square it, then apply unary negation to the result.” If you write (-3)^2 then it is shorthand for “take the number (-3) and square it.” Two different operations.
(-3^2)
Parentheses: Okay, do the operation in parentheses first. (This rule is actually irrelevant because we have only in-parentheses operations.) This leaves us, basically, with -1*3^2 as friedo said.
Exponents: Process the exponentiation. -1 * 9.
Multiplication: -9.
And we’re done.
We’d have a different result if the problem were (-3)^2, in which case we’d process the parenthetical operation first, then the exponentiation to get 9.
Does this explanation make sense to you, Jinx?
Folks, I think Jinx has asked a sophisticated question with subtleties that most kids that have finished high-school math with straight A’s can’t answer.
I’m trying to come up with a plain English answer that doesn’t rely on mechanical rules and conveys the conceptual ideas. Maybe I’m wrong but I think that’s what Jinx is looking for. I’m sure the other math geniuses will beat me to it.
The number -3 is perfectly valid as a number. The problem is that when you write “-3^2” there is an ambiguity: do you carry out the negation or the exponent first? Without a generally agreed convention, you could do it either way, but people have agreed on a convention, as said above.
To avoid the ambiguity, especially if you are not sure if your readers will know the convention, you can write it -(3^2) or (-3)^2.
I don’t understand, “In other words, the 3 to the left of the zero is not unique from the 3 to the right of zero.” Just because the same digit is used in the usual expression does not mean that -3 = 3. You might as well argue that 12 = 21 because they use the same digits, or that 1-2 = 2-1.
The book is right, and there IS such a number as -3. There is no problem with the concept of -3 as a real number. The issue here is that the number -3 does not appear anywhere in the expression -3[sup]2[/sup] which by convention means -(3[sup]2[/sup]).
Do not confuse the concept of a number with how to name that number. There are, of course, negative numbers, those to the left of zero on the number line. But there is no atomic way to name those numbers. The way we name them is to apply unary negation to the positive number. The result is the conceptual negative number. The term
-3
is not an atom, it is an expression. It indicates an operation that evaluates to the number that we think of as “negative three.” (“Unary” means that we are applying an operation to a single operand; most operations take two operands.)
ETA: When we add in other operations, we have to consider precedence. Unary negation takes a high precedence, but exponents take a higher precedence. It is unfortunate that we do not teach Please Excuse Urine on My Dear Aunt Sally 
I’m not sure it’s that complicated. I think, and I may be wrong, that the issue is familiarity with the notational conventions that are expressed in the mechanical rules.
The problem with the problem is that it’s ambiguous to say the least. The book may be technically correct, but most people would simply use -(3^2) to indicate an answer of -9. To me, it’s way too easy to assume (-3^2) means (-3)^2 and hence give an answer of 9. And teachers wonder why so many kids hate math…
First, I must say I was never taught the math jargon “unary”. Next, for math to be consistent with itself, there is no such quantity of -3 as said above it is a combination. Therefore, the only correct way to express a negative quanity, such as negatives on a numberline, would be (-3). And since -3^2 is treated differently than (-3)^2, then one can only conclude that -3 <> (-3). Now, the math folks will argue to the contrary, but rather I’d say their notation is flawed! They have to make up their minds! If indeed -3 = (-3), then so it IS always equal forever and ever…and by that logic, therefore: -3^2 = (-3)^2 = 9. Ha!
I didn’t interpret the question that way.
As an example, I think CookingWithGas’s wording of his answer “the term -3 is not an atom” conveys deeper meaning and is not apparently obvious even if you’ve thoroughly memorized “please-excuse-my-dear-aunt-sally”.
Exactly.
To confuse things further, when writing out an equation it is quite common to have more whitespace between elements rather than have loads of brackets. For example:
At school (in the UK) we used to sum things of with BODMAS:
Brackets
Or
Divide
Multiply
Add then
Subtract
When it comes down to it, if it doubt then add brackets.
It seems logical enough to me.
-x[sup]2[/sup] = -4
x[sup]2[/sup] = 4 (multiply both by -1)
x = 2
-2[sup]2[/sup] = -4
I suppose you can make two different rules, depending on whether you’re squaring a variable or a literal, but why?
This is pretty much your answer; it’s a notational problem. (“Flawed” may be too strong, since the answer is well-defined.)