['here are several algoritluns or methods of computing the square-\"cube root of a nonnegative real lumber S; in. we?" it? One of the rst algorithms used for approximating it? or w is the Eabylcnian method= or "Hero's method"= named aer the rstcentury Greek mathematician Hero if Alexandria who gave the rst explicit description of the method. n this method nding it? or 3 S is the same as} respectively} solving the equation HI) = x2 I: {l or f(x) = x3 S = CI for a positive x. Therefore} an},r general numerical rootnding ngorithm can he used and Newton's method= for example, is one of them. Newton's Method Jrovides a convergent numerical approxmlation for nding roots: with the caveat that an initial guess must be chosen. fou will use Neuton's Meaod as follows: 1. Begin with an arbitrary positive starting value Xa (the closer to the actual square root of S} the better}. In our case: the initial guess will be chosen according to the following formula 5 In = E 2. LIE-'t xi+1 = x1- fli} f'txz']