The principal ''n''-th root <math>\sqrt[n]{A}</math> of a [[negative and positive numbers|positive]] [[real number]] A, is the positive real solution of the equation
:<math>x^n = A</math>
(Forfor integer ''n'' there are ''n'' distinct [[complex number|complex]] solutions to this equation if <math>A > 0</math>, but only one is positive).
There is a very fast-[[convergence|converging]] '''n-th root algorithm''' for finding <math>\sqrt[n]{A}</math>:
#Repeat step 2 until the desired precision is reached.
ThisA isspecial acase generalization ofis the well-knownfamiliar [[Square_root#Square_roots_using_Newton_iteration|square-root algorithm]]. By setting ''n'' = 2, the ''iteration rule'' in step 2 becomes the more familiar square root iteration rule:
Several different derivations of this algorithm are possible. One [[N-th root algorithm#Derivation from Newton's method|derivation]] shows it is a special case of [[Newton's method]] (also called the Newton-Raphson method) for finding zeros of a function <math>f(x)</math> beginning with an initial guess. Although Newton's method is iterative, meaning it approaches the solution through a series of increasingly-accurate guesses, it converges very quickly. The rate of convergence is quadratic, meaning roughly that the number of bits of accuracy doubles on each iteration (so improving a guess from 1 bit to 64 bits of precision requires only 6 iterations). For this reason, this algorithm is often used in computers as a very fast method to calculate square roots.
For large ''n'', the ''n''-th root algorithm is somewhat less efficient since it requires the computation of <math>x_k^n</math> at each step, but can be efficiently implemented with a good exponentiation algorithm.
