If $x^2$ and $x^3$ are rational, does it imply that $x$ is rational?

Question

It is given that $x^2$ is rational and $x^3$ is rational.

Is $x$ rational for all cases satisfying these conditions or is there are case where $x$ won't be rational? If so, then what other condition(s) are required?

Answer 1

Yes, indeed it implies that $x$ is rational. Because $x^2$ and $x^3$ are rational, $\frac{x^3}{x^2}$ is also a rational number if $x≠0$. ($\Bbb Q \setminus \{0\}$ is a group under ordinary multiplication). Hence $x$ is a rational number.