  Finite discrete-time dynamical systems (FDDS) model phenomena that evolve
deterministically in discrete time. It is possible to define sum and product
operations on these systems (disjoint union and direct product, respectively)
giving a commutative semiring. This algebraic structure led to several works
employing polynomial equations to model hypotheses on phenomena modelled using
FDDS. To solve these equations, algorithms for performing the division and
computing $k$-th roots are needed. In this paper, we propose two polynomial
algorithms for these tasks, under the condition that the result is a connected
FDDS. This ultimately leads to an efficient solution to equations of the type
$AX^k=B$ for connected $X$. These results are some of the important final steps
for solving more general polynomial equations on FDDS.
