Irrationality of ∛2

Assume $\sqrt[3]{2}=p/q$ where $p$ and $q$ are positive integers such that $\gcd(p,q)=1$. Then $p^3=2q^3$ and $P$ is even; write $p = 2p'$ where $p'$ is a positive integer. That implies $2 q^3 = p^3 = 2^3 p'^3$, hence $q^n = 2^{n-1} p'^3$, and thus $q$ is even. This is impossible since $\gcd(p,q)=1$.

Conclusion. $\sqrt[3]{2}$ is irrational.