Let \( K \) be a number field, and let \( \ell \) be a prime number. Fix some elements \( \alpha_1,\ldots,\alpha_r \) of \( K^\times \) which generate a subgroup of \( K^\times \) of rank \( r \). Let \( n_1,\ldots,n_r, m \) be positive integers with \( m \geq n_i \) for every \( i \). We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension \( K(\zeta_{\ell^m}, \sqrt[\ell^{n_1}]{\alpha_1},\ldots,\sqrt[\ell^{n_r}]{\alpha_r}) \) over \( K(\zeta_{\ell^m}) \) for all \( n_1,\ldots,n_r,m \). This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.