When considering the formula for the solution to a degree three polynomial equation, I observe that it involves two different square roots inside of two different third roots. From the standpoint of complex analysis, each square root represents two different possible branches, and each third root represents three different possible branches. So in principle the cubic formula appears to give possibly $3\cdot2\cdot3\cdot2=36$ different roots. Of course we know that a degree three polynomial equation really has only three solutions.
Without using this latter fact, working just off of the formula, is there a way to see that the set of all possible choices of the roots found in the cubic equation reduces to just three numbers?