Completely factorise the given number. Then group its factors into sets of 3. Take just one representative from each set and multiply these together. The answer is the required cube root.
For example,
1728 = 2*2*2*2*2*2*3*3*3 = (2*2*2)*(2*2*2)*(3*3*3)
So, the cuberoot is = 2 * 2 * 3 = 12
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Perfect cubes have prime factors in multiples of 3.
3 x 3 x 3 = 27
2 x 2 x 2 x 5 x 5 x 5 = 1000
a3 + b3 = (a + b)*(a2 - ab + b2) which gives an initial factorisation as shown in the two brackets. Each of these may have further factors.
78
24 and 16 are factors of 48
8 and 5
a3 + b3 = (a + b)(a2 - ab + b2) a3 - b3 = (a - b)(a2 + ab + b2)
It is important to determine what the correlation is so that you can control it. If you can find out how two factors are related you can manipulate the situation.