Distinct Roots (not that kind)

Define $M_i$ as the minimum positive integer such that all elements in the set $S = \left\{M_i, \left\lfloor\sqrt[2]{M_i}\right\rfloor, ..., \left\lfloor\sqrt[i]{M_i}\right\rfloor\right\}$ are pairwise distinct. What are the last three digits of $\sum _{i=2}^{10} M_i$ ?