Are there any rational numbers $x, y, z$ with $xyz \neq 0$ and coprime numerators such that $x^3 +y^3 = z^4$ ?



On the equation $x^3 + y^3 = z^4$
It is a rational surface.


One easy parametrization is
$x=s^4 + s\; t^3$, $y=s^3\;t + t^4$ and $z=s^3 + t^3$. From this you should be able to find as many examples as you like.


