How do you find the twelfth root of a number?

Answer 1

You raise the number to the power 1/12. For example, in Excel, to take the 12th root of 2, you can type:=2^(1/12)

You can also try raising different numbers to the 12th power, until you get close enough. There are more efficient ways to do this; I understand that calculators and computers use formulae based on the exponential and natural logarithm function.

Answer 2

The answer depends on the options available to you. Clearly, the answer is trivial if you have a scientific calculator. The answer is x^(1/12).

If you have a non-scientific calculator which does do square and cube roots, then the answer is

sqrt(sqrt(cuberoot(x))).

Otherwise you will need to use a numerical method which will require a fair bit of number crunching. One such is the Newton-Raphson method. To find the 12th root of k, this method requires you to solve x^12 - k = 0.

Start with a crude estimate, x(0).

The use the iteration:

x(n+1) = x(n) - f[x(n)]/f'[x(n)]

where f[x(n)] = [x(n)]^12 - 12

and

f'[x(n)] = 12*[x(n)]^11