Is it possible to use the root function to attack RSA?
If root is just the function from calculus, then

at most ceil(root(n,3)) elements cleartext of ​ {0,1,2,3,...,n-2,n-1}

are such that root(cleartext,3) is an integer
and
when its output is not an integer, that output is not a valid signature,

since valid signatures must be elements of ​ {0,1,2,3,...,n-2,n-1}

.

If root is modular, then for carefully chosen large moduli n, there's no publicly known 

feasible classical way to have a non-negligible probability of computing root.

However, quantum computers change things quite significantly.