AbstractThis note gives a general 2nd order solution, with the corresponding 3rd order correction, by means of which a good numerical answer can be obtained from an approximate estimate in one operation, and results of high accuracy by one or two iterations. The method is applied to integral roots of numbers, and to common types of equations.For the extraction of square roots and cube roots of high accuracy, the general method is developed in the form of series designed to simplify the computation, and one example is given in full with an accuracy of over 30 significant figures.