Step-by-step answer:
You can use Newton's method as follows:
It is an iterative method, meaning that from an approximation (i.e. an approximate solution), we can refine the answer to get a closer approximation. By repeating the process (iteration), we can get an answer as accurate as we wish.
The basis of the formula is based on
x1=x0-f(x0)/f'(x0)
where
x0 is a given approximation
x1 is a better approximation
f(x) is a given function (for which we need the roots)
f'(x) is the derivative of the function.
Skipping all details and applying directly to find the cube root, we have
N=the number for which the cube root is desired
f(x)=x^3-N
f'(x)=3x^2
and x0 is an initial approximation that we need to provide (from the integer cubes, for example).
Say we need to find the cube-root of 124.
We know that 5^3 = 125, a rather close approximation, but the cube root is slightly less than 5.
To find a better approximation, we apply Newton's method, and calculate mentally:
x0=5, N=124
x1 = x0-(x0^3-124)/(3(x0^2)) [next substitute values]
= 5 - (125-124)/(3(5^2)) [ next simplify ]
= 5-1/75 [next, rearrange to calculate mentally ]
=5-(1/25)/3 [ next, substitute 1/25 = 0.04, 1/3 division can be done mentally ]
= 5 - 0.04/3 [ divide 0.04/3 mentally ]
= 5 - 0.013333 [ subtract ]
= 4.98667
(exact value = 4.98663, first approximation is already quite close)