Compact relationships between invariants of the unitary groups $U(n)$ and power sums

Compact invertible relationships between power sums and the eigenvalues of the invariants of two types of the unitary groups $U(n)$ are derived. Theysimplify calculations, particularly at large $n$, compared with the well-known studies. The connection in these expressions with Bell polynomials and Bernoulli numbers is demonstrated. Examples of the use of the obtained expressions in various models of nuclear physics are given.