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

  • Thread starter ironman1478
  • Start date
  • #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


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

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
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

  • #4
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:
  • #5
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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