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

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

Homework Statement


Given x>0 and y>0, show that x^3 = y^3 => x = y. Does this hold for all every x and y?


Homework Equations


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


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:

Answers and Replies

  • #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?
 
  • #3
edit: nvm, i read the question wrong. i see what you and the question mean now lol

thanx
 
  • #4
I like Serena
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if y is negative and x is positive then their cubes can't be equal right?
No....? :confused:

But then the conditions do not hold either:
y>0
x^3=y^3
x=y
 
  • #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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