root question

Holocene

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?

Count Iblis

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

slider142

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

Count Iblis

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.

Holocene

wow, that's pretty neat. Thanks!

Count Iblis

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.