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Show that x^3=y^3 implies x=y

  1. Nov 9, 2011 #1
    the question is from the book "elementary geometry from an advanced standpoint 3rd edition" by edwin e. moise

    1. The problem statement, all variables and given/known data
    Given x>0 and y>0, show that x^3 = y^3 => x = y. Does this hold for all every x and y?


    2. Relevant equations
    a^3-b^3=(a-b)(a^2+ab+b^2)=0


    3. The attempt at a solution
    so what i did was subtract y^3 from both sides to get
    x^3-y^3 = 0

    then i factored it out to
    (x-y)(x^2+xy+y^2) = 0

    because we know that x>0 and y>0, the second term (x^2+xy+y^2) is always positive. because of this (x-y) must equal zero
    then we setup the equation x-y=0
    x=y.

    i think i did this correctly, but since i am teaching myself out of this book (i just want to learn more about geometry because i felt like i was never taught it well) i have no way of verifying if this is correct
     
    Last edited: Nov 9, 2011
  2. jcsd
  3. Nov 9, 2011 #2

    I like Serena

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    Welcome to PF, ironman14781 :smile:

    Looks good!

    Btw, this holds true for every x and y (real numbers).
    Why would you think otherwise?
     
  4. Nov 9, 2011 #3
    edit: nvm, i read the question wrong. i see what you and the question mean now lol

    thanx
     
  5. Nov 9, 2011 #4

    I like Serena

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    No....? :confused:

    But then the conditions do not hold either:
    y>0
    x^3=y^3
    x=y
     
  6. Nov 10, 2011 #5

    HallsofIvy

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    It is not a matter of "x< 0, y> 0". If x and y are any two numbers such that [itex]x^3= y^3[/itex], then x= y. It may be that x and y are both postive or that they are both negative (or both 0).
    If x and y are both negative then xy is positive so it is still true that [itex]x^2+ xy+ y^2[/itex] is positive.
     
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