Cube Root AlgorithmDate: 04/04/97 at 12:37:41 From: Andrew Walters Subject: Cube Root Algorithm Hi, Dr. Math! Is there an algorithm for working out the cube root of numbers without a calculator? My teacher said there was, but told me to research the answer. Thanks for your help, Andrew Date: 04/04/97 at 15:48:27 From: Doctor Anthony Subject: Re: Cube Root Algorithm Cubic equations can be solved without recourse to approximate methods, but they can be very difficult and involve the use of complex numbers. Judging by your question, I would guess that you have not yet met complex numbers (they make use of sqrt(-1)= i). For the time being you would need to stick to approximate methods. If you had to solve x^3 = 86, you could proceed by Newton's Method as follows: f(x) = x^3 - 86 = 0 f'(x) = 3x^2 Then we can use an iterative formula to improve on the accuracy of the root. If x0 is our first guess at the root, then a better approximation is given by: x1 = x0 - f(x0)/f'(x0) x1 = x0 - (x0^3-86)/(3x0^2) Now 4^3 = 64 and 5^3 = 125, so we know the root lies between 4 and 5, closer to 4. So let us start with x = 4.3 x1 = 4.3 - (4.3^3 - 86)/[3(4.3)^2] = 4.417 x2 = 4.414 x3 = 4.414005 and x4 = 4.414004962 If you use a calculator to work out cube root(86) you get the same value as x4, so the method gets the right answer quite quickly. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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