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Root questionby Holocene
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#1
Nov2107, 07:32 PM

P: 231

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? 


#2
Nov2107, 08:05 PM

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:
[tex]\left(1\frac{1}{n}\right)X + \frac{Y}{n X^{n1}}[/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 


#3
Nov2107, 08:12 PM

P: 898

If you want to know the general theory behind the above method, see Newton's method.



#4
Nov2107, 08:13 PM

P: 2,157

Root question
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.



#6
Nov2107, 08:44 PM

P: 2,157

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. 


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