# -x2 and (-x)2

#### confused14?

##### New member
considering these expressions -x2 and (-x)2 are they the same? i'm lost. please expain :?: :?:

#### yoda

##### New member
are the 2s exponents?

#### Mrspi

##### Senior Member
confused14? said:
considering these expressions -x2 and (-x)2 are they the same? i'm lost. please expain :?: :?:
-x<SUP>2</SUP> and (-x)<SUP>2</SUP> are not the same. Here's why:

Remember the order of operations? First, you do any operations inside parentheses. Next, you do powers or roots (exponents).

-x<SUP>2</SUP> has no operations in parentheses. So, you start by doing the squaring. THEN you find the opposite of the result.

For example, if x = 3, -x<SUP>2</SUP> would become -3<SUP>2</SUP> Start by squaring 3 to get 9. Then, find the opposite of 9, which is -9. So -3<SUP>2</SUP> = -9.

Or, if x = -5, -x<SUP>2</SUP> would become -(-5)<SUP>2</SUP>. Multiply (-5)(-5) to do the squaring. (-5)(-5) is 25. Then, find the opposite of 25, which is -25.

(-x)<SUP>2</SUP> DOES have an operation inside the parentheses....it says "take the opposite of whatever x is." You do that first. THEN you do the squaring.

For example, if x = 3, then (-x)<SUP>2</SUP> becomes (-3)<SUP>2</SUP>. The opposite of 3 is -3, and when you square -3, you do (-3)(-3) and get 9.

Or, if x = -5, then (-x)<SUP>2</SUP> becomes [-(-5)]<SUP>2</SUP>. The opposite of -5 is 5....now you have (5)<SUP>2</SUP>, or 25.

For any real, non-zero value of x, -x<SUP>2</SUP> is always negative. And (-x)<SUP>2</SUP> is always positive.

#### soroban

##### Elite Member
Hello, confused!

Considering these expressions $$\displaystyle \,-x^2$$ and $$\displaystyle (-x)^2$$
Are they the same? . . . . no

This is a source of expected confusion in math classes.

I have a number of approaches to explaining the difference.
I'll try one of them now . . .

We know exponents affect only the factor immediately to its lower-left.

Example: $$\displaystyle \:3ab^2$$

We know that only the $$\displaystyle b$$ is squared; it is actually: $$\displaystyle \:3\,\cdot\,a\,\cdot\,b\,\cdot\,b$$

If they wanted the $$\displaystyle a$$ to be squared too, they must say so.
. . They should write: $$\displaystyle \:3(ab)^2$$

If the 3 is included in the square, it should be written: $$\displaystyle \ 3ab)^2$$

Example: $$\displaystyle \,-a^2$$
We can see that the $$\displaystyle a$$ is squared . . . but is the "minus" also squared?

No . . . The problem is actually: $$\displaystyle \,-1\,\cdot\,a^2$$, you see.
. . If the "minus" is to be squared, it would be written: $$\displaystyle \ -a)^2$$

You may well ask "What in blazes is that minus doing in front?"

Answer: It's doing what it always does . . . changing the sign of the quantity.

Just as$$\displaystyle \,-4$$ tell us that we have a "negative 4",
. . $$\displaystyle \,-x$$ means "the negative of $$\displaystyle x$$"
. . and$$\displaystyle \,-a^2$$ means "the negative of a-squared".

If you are given $$\displaystyle \:16\,-\,3^2$$, you would get: $$\displaystyle \:16\,-\,9\:=\:7$$

If you are given$$\displaystyle \,-3^2\,+\,16$$, I hope you would get the same answer.