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The N'th Root

Date: 07/26/97 at 21:02:25
From: Saxby Brown
Subject: How to find the n'th root without a calculator

I understand how to get the square root of a rational number without a 
calculator, and I am wondering if there is any way you can generalize 
that algorithm to work for the n'th root of any number?

Date: 07/27/97 at 14:26:48
From: Doctor Jerry
Subject: Re: How to find the n'th root without a calculator

Hi Saxby,

The divide-and-average algorithm for approximating the square root of 
a number came from geometry and is a special case of Newton's method, 
applied to the function f(x) = x^2 - a, where a>0 is the number whose 
square root is wanted.

If Newton's method is applied to f(x) = x^n - a, the resulting 
algorithm gives an arithmetic-based method for calculating the nth 
root of a positive number.  Let x_1 (x sub 1) be the first guess; 
then x_2, x_3, x_4,..., are the successive approximations, where the 
m+1 st approximation is related to the mth by the formula

   x_{m+1} = [(n-1)/n]*x_m + a/[n*(x_m)^{n-1}].

So, if n = 5, a = 2, and x_1=1.5,

   x_2 = 1.27901234568
   x_2 = 1.17268228941
   x_3 = 1.14965954326
   x_4 = 1.14869996088

The 5th root of 2, by calculator, is 1.148698355

-Doctor Jerry,  The Math Forum
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