# Debate that -2^2=-4

## Main Question or Discussion Point

Ever since Algebra, I've been taught that -2^2=4
On another message forum, however there was a debate that -2^2=-4.
They came up with the conclusion that:
-2^2 is the same as -1(2^2) which equals -1(4) equaling -4.
I just wonder what the correct answer is.

It depends on your parentheses. Normally, -22 means -(22), which is -4. But if you mean (-2)2, then that is +4.

enigma
Staff Emeritus
Gold Member
Originally posted by gcn_zelda
Ever since Algebra, I've been taught that -2^2=4
On another message forum, however there was a debate that -2^2=-4.
They came up with the conclusion that:
-2^2 is the same as -1(2^2) which equals -1(4) equaling -4.
I just wonder what the correct answer is.
(-2)^2 = 4

and

-(2^2) = -4

are not the same thing.

ShawnD
2^2 is the same as -1(2^2)
That is an incorrect assumption. You can't just start factoring numbers and taking them out of bracks with exponents, it just doens't work that way. It's hard to put into words so just look at the example.
Lets say I want to square root A^2 + 2AB + B^2. I would write that as (A^2 + 2AB + B^2)^(1/2). The answer to this is (A + B) of course. Now lets say I used the silly logic mentioned above, i factor out an A + B. Now I end up with (A + B)(A + B)^(1/2) which I can't even give an answer for.
Do you see how factoring out completely changes the answer? That is why we never do that.

(-2)^2 = 4
(-2i)^2 = -4

If you want that negative to stay, you need to put an i there :D

(-2)^2 = (-1)^2*(2)^2 = 1*(2)^2 = 2^2 = 4.

If you take the -1 factor out of -2 you have to raise it to the power of 2 as well.

Ever since Algebra, I've been taught that -2^2=4
Then, I think, you have been taught wrongly. Or you remember wrongly.

I say:
-22 = -4.

Because convention is: Power has priority over multiplication.

-2 = -1*2.
So,
-22 = -1*22.
Since power has priority, this equals
-1*4 = -4.

Like Ambitwistor and enigma pointed out, -2^2 is not -1*2^2.

-2 = -1*2
In the next step, squaring both sides results in:
(-2)^2 = (-1*2)^2
In your calculations, you failed to square the -1. While exponents are evaluated before products, paretheses are resolved before exponents. Therefore -1*2 is evaluated first, as -2, which is then squared, resulting in 4.

Another way of evaluating the expression is to apply the exponent to both terms on the right side:
(-1)^2 * (2)^2
Which is equal to:
1 * 4

Hurkyl
Staff Emeritus
Gold Member
Like ambitwistor said, it depends on where you mean the parentheses to be.

However, the standard is that

$$-a^b = -(a^b)$$

so without any context to indicate otherwise, any mathematician would unambiguously interpret it as above.

worl sqrt(-9) be 3i or 9i???

sqrt of (-9):
sqrt of (-9) = sqrt of (9) times the sqrt of (-1)
so sqrt of (9) = 3 and sqrt of (-1) equals i
then sqrt of (-9) = 3i
:)
jk