How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational? [closed]

Please see this question's title.

Suppose $p^{1/n}=\frac{a}{b}$, a fraction in its lowest terms. Then $pb^n=a^n$, so $p$ divides $a^n$, so $p$ divides $a$, so $p^n$ divides $a^n$. This means $p$ also divides $b$, a contradiction.

Hint $\ $ By the rational root test, any rational root of $\rm\ x^n - a\ $ is integral, so every prime in the unique prime factorization of $\rm\ a\ $ occurs to a power divisible by $\rm\:n.$

You've added the number theory tag but I'm trying to solve it using ring theory.

We know if $p$ is a prime then $n>1$ implies $x^n-p \in \mathbb{Z} [x]$ is irreducible. Hence it is irreducible in $\mathbb{Q} [x]$. In particular, it has no rational roots.