-x2 and (-x)2

confused14?

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Nov 13, 2006
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considering these expressions -x2 and (-x)2 are they the same? i'm lost. please expain :?: :?:
 

yoda

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Nov 12, 2006
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are the 2s exponents?
 

Mrspi

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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

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Jan 28, 2005
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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.

 
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