assume it is rational. then we can write for , .
we square both sides to obtain:
now the right side of the equation is a rational number (by our assumption). thus we must have that is rational. therefore, we can prove the claim by showing that this is false.
now proving is irrational is also done by contradiction, and usually done the same way is proven to be irrational (see here)
but there is a shorter way.
consider the equation
by the rational roots theorem, if rational roots exist, they will be one of . but none of these solve the equation. however, solves the equation and is therefore not a rational number.
so seeing that is not rational, we have our contradiction