Is there any "pencil and paper" method to find the nth root of a number? Since multiplying a number by itself any number of times quickly yeilds extremely large numbers, trial and error might seem to pinpoint the root of a number, so long as it is a perfect square or cube or whatever. But, is there any real way to pinpoint the root of a number without using a calculator or trial and error?
To compute the nth root of a number Y, just make some guess X and then improve that guess using the formula: [tex]\left(1-\frac{1}{n}\right)X + \frac{Y}{n X^{n-1}}[/tex] 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
So, this is still trial and error, but it converges very fast. At each step you double to correct number of digits. You go from a wild guess to a number that is correct to ten significant digits in about four iterations.
The case n = -1 is also very useful. In that case X = 1/Y but Newton's method gives: [tex]2X - X^{2} Y[/tex] Since there are no divisions in here, you can use it to do divisions. It's much faster than long division.