Try using the Rational Root Theorem :-)
Cube root of 5 is a root. Find all the possible rational roots. If they don't satisfy the equation, then there are no rational roots => cube root of 5 is irrational.
If you are not familiar with the rational-root theorem, you can still tackle the problem directly.
Assume that is rational, so where and .
If you then plug this back into the equation, you would find that 5 divides as well, contradicting the assumption that and are coprime.