The N'th RootDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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