Prove that $2^{1/n}$ is irrational Proof by contradiction, Assume $2^{1/n}$ is rational so:
$$2^{1/n} = \frac ab $$
where a,b have no common factors.
$$2 = \frac{a^n}{b^n}$$
$2$ divides LHS, therefore $2$ divides RHS
so $2$ divides $a^n$ or $2$ divides $b^n$ which implies $2$ divides $a$ or $2$ divides $b$. 
Stuck on what to do next.

Factor $a$ and $b$ into products of primes. 
We have the identity $2b^n = a^n$; compare the exponents of the primes on both sides of the equation (and look in particular at the exponent of 2).
