Can the cube of 2 different complex numbers be the same?

Question

Can the cube of 2 different complex numbers be the same?

I think it cannot be the same, but I don't really know how to prove it. I tried to expand it but it gives a very ugly result.

Answer 1

Assuming you have to ask, you are yet to study De Moivre's formula (in particular, see the applications section). For any complex number $z_1$ and integer $n$, you can find $n-1$ other complex numbers $\{z_2, z_3, \ldots , z_n\}$ that will all have the same $n$-th power as $z_1$ (also, the same absolute value). Pretty neat, eh?

Also, if you try plotting those numbers in the complex plane, they will all be be on a circle with a radius equal to the absolute value of any of those complex numbers. Complex numbers are cool.

Answer 2

If you take $x,y\in\mathbb{C}$ and $x^3=y^3$, we get $$ x^3-y^3=(x-y)(x^2+xy+y^2)=0 \Rightarrow x=y , x^2+y^2+xy=0 $$ Solving the second equation you get $y=\omega x$ or $y=\omega^2 x$. You can check $y^3=\omega^3x^3=x^3$ etc.