Proving x=y when x^n=y^n for odd n

[Antonio94]

If x^n=y^n , n= odd
Then x=y

I know that an odd is n=2k+1

So x^(2k+1)=y^(2k+1)

I don't know how to finish it. Please help.

 

[haruspex]

You are necessarily dealing with reals for this result to be true, so you can break it into cases according to x < y, |x| < |y| etc. Try induction.

 

[slider142]

If this is from Spivak's Calculus, there is an alternative approach that avoids induction (as that property is first introduced in the next chapter). In particular, I used the fact that the contrapositive of a conditional statement has the same truth value as the original statement. Therefore, if you can prove that the contrapositive of the statement is true, you have proven the truth of the original statement. The contrapositive of this statement is particularly simple, as long as you write it properly.

 

[Gzyousikai]

You have too assume that $$x, y$$ are reals first...

divide y both side, and let $$\frac{x^n}{y^n}=t$$ deduct to proof $$t^n=1$$ has only one real solution which is one. which is equivalent to $$(t-1)\left(\sum_{i=0}^{n}t^i\right)=0$$ which is relative to the root of unity.