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Confusion about -x^n

  1. Nov 15, 2013 #1
    Hello,

    This is my first post here. I am sorry if this is the wrong place for this question or if my phrasing is unclear.

    -x^n≠(-x)^n
    -x^n=-(x^2)

    I am slightly confused by this because at the level I am at right now (i.e. pre-calculus) and before, I have seen statements that contradict this. For example, I feel like I have been told from beginning algebra until now that -2^2=4. When is this true and when is it not?

    I suppose it is a matter of notation?

    Thank you.
     
  2. jcsd
  3. Nov 15, 2013 #2

    FeDeX_LaTeX

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    Gold Member

    Depends on how you place the parentheses.

    ##(-2)^2 = (-2) \cdot (-2) = 4##
    ##-(2^2) = -[(2) \cdot (2)] = -4##

    In general, ##-(x^n) = (-x)^n## if n is an odd integer.
     
  4. Nov 15, 2013 #3
    Thank you for your swift response.

    If there are no parentheses, what is assumed? Or does that vary from text to text?
     
  5. Nov 15, 2013 #4

    FeDeX_LaTeX

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    Gold Member

    I'd say that if there are no parentheses given, it's safe to assume that they mean ##-x^n = -(x^n)##, unless they explicitly say otherwise.
     
  6. Nov 15, 2013 #5

    Mark44

    Staff: Mentor

    Probably a typo in the 2nd equation. I guess you mean -x^n = -(x^n).
    You should not have been told that -22 = 4. You might be misremembering what was said in class.
    I edited my earlier response. What you're saying is correct, FedEx_LaTeX, but you don't need the qualifier. ##-x^n## and ##-(x^n)## mean exactly the same thing.

    According to the operator precedence, exponents are evaluated before signs.
    -xn means the opposite of xn, or -(xn). If you want to raise -x to a power you have to include parentheses around the thing being raised to the power, like this: (-x)n.
     
    Last edited: Nov 15, 2013
  7. Nov 15, 2013 #6
    In addition, be careful about the first statement that you made.
    -xn≠(-x)n does not hold for all n and x that you are used to using.
    This statement is FALSE for all odd n. There is also some x for which this is false.
     
  8. Nov 15, 2013 #7
    Mark, I believe what you've written agrees with what FeDeX wrote.
     
  9. Nov 15, 2013 #8

    Mark44

    Staff: Mentor

    I now see that, and have revised what I wrote in that thread. My only excuse is that the "unless they explicitly say otherwise" part threw me off.

    I'm so used to seeing people write, for example, -22 = +4, that I thought (erroneously, mea culpa) that that's what FedEx_LaTeX was saying. Hopefully, things are clear now.
     
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