Question

how do you find the cube root of a number?

Answer 1

Your subject is not geometry.

These days, almost everyone uses a pocket calculator for roots. A table of logarithms or a slide rule can also be used, but no one does that anymore.

Since you are using a computer, you can also raise the number to the 1/3 power, using Google. For example: for the cube root of 27, write in the search box
27^(1/3) =
and you should get the answer, 3.

Answer 2

suppose you want the cube root of k

take a reasonable guess, call it x
sub that guess into (2x^3+k)/(3x^2)

if the result differs from x, make the result your new x and repeat the process.

After a few loops of this process, your result will get closer to your input of x.
This is called Newton's Method

e.g. find cube root of 30, so k=30
my starting guess is x=3

guess -- result
3 .....3.11111111
3.1111111 3.10723734
3.10723734 3.107232505
3.107232505 3.107232506

notice for all practical purposes, my input is equal to my output after only about 4 "iterations" .
The closer your original guess is to the actual cube root, the less steps it will take.

my calculator says, cube root of 30 = 3.107232506

I know this is a rather tedious process, but it involves only the basic operations of addition, multiplication and division, and you did ask for a method of finding cube roots.

Answer 3

If you don't even have a calculator, (as in an exam, or while you're camping), you can use the following method (algorithm):
http://www.mathpath.org/Algor/cuberoot/algor.cube.root.htm

You will need pen and paper though.