# Root question

by Holocene
Tags: root
 P: 2,157 To compute the nth root of a number Y, just make some guess X and then improve that guess using the formula: $$\left(1-\frac{1}{n}\right)X + \frac{Y}{n X^{n-1}}$$ You can iterate this to make further improvements. E.g. suppose you want to estimate the cube root of 10. Then you can take X = 2. the formula gives: 4/3 + 10/(3*4) = 2 + 1/6 If you then take X = 2+1/6 and insert that in the formula to get 2.1545. Iterating again gives 2.15443469224. Now, believe it or not but: 2.15443469224^3 = 10.0000000307
 Quote by Count Iblis To compute the nth root of a number Y, just make some guess X and then improve that guess using the formula: $$\left(1-\frac{1}{n}\right)X + \frac{Y}{n X^{n-1}}$$ You can iterate this to make further improvements. E.g. suppose you want to estimate the cube root of 10. Then you can take X = 2. the formula gives: 4/3 + 10/(3*4) = 2 + 1/6 If you then take X = 2+1/6 and insert that in the formula to get 2.1545. Iterating again gives 2.15443469224. Now, believe it or not but: 2.15443469224^3 = 10.0000000307
 P: 2,157 The case n = -1 is also very useful. In that case X = 1/Y but Newton's method gives: $$2X - X^{2} Y$$ Since there are no divisions in here, you can use it to do divisions. It's much faster than long division.