In this paper, we study the principal specialization of monomial symmetric polynomials and investigate the special values of these polynomials at \[ ζ_{(n,k)} := ( 1, ζ_n, ζ_n^2, \dots, ζ_n^{kn-1} ), \] where $ζ_n$ is a primitive $n$th root of unity. We give explicit formulas for several classes of special values. We also show that these special values naturally appear as the coefficients in the expansion of the $k$th power of the circulant determinant of order $n$ (the group determinant of the cyclic group of order $n$). These results extend Ore's formulas for the case $k = 1$. Furthermore, we determine the number of terms in the $k$th power of the group permanent of the cyclic group of order $n$. This extends Brualdi and Newman's result for $k = 1$.