Summary: An iterative algorithm for finding $\root n \of {m^u}$, ($m > 0$, $u < n$), is developed which involves generating a sequence of approximations to $\root n \of {m^u}$ using the concept of eigenvectors. The convergence of this method is then established by studying the eigenvalues and eigenvectors of a matrix $A_{n}$, directly related to the algorithm itself. The matrix $A_{n}$ is constructed using the eigenvalues and eigenvectors, applying the concepts of diagonalization. An algorithm for finding higher powers of $A_{n}$ is explained. Using these higher powers of $A_{n}$, a direct method is also derived. Two numerical examples explaining the methods are given.