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You have to know the squares, cubes, whatever before you can start deriving roots.
Using your example, we know the cube root of 27 is 3, but suppose you asked what is the cube root of 28?
Well, we start by finding the nearest lower perfect cube, which is 27, and its root, which is 3.
So the first number of the cube root of 28 is 3, with a remander of 1.
next, what is the next lowest cube of 10 (1.0)? That is 8 or root 2, remainder of 2. (we are now at 3.2).
now what is the next lowest cube of 20 (.20)? That is still 8 or root 2, remainder 12 (we are now to 3.22)
what is the lowest cube of 120 (.120)? That is 64 or root 4, remainder 56 (we are now at 3.224)
and so on until you get the accuracy you desire.
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"Of all tyrannies, a tyranny exercised for the good of its victims may be the most oppressive. It may be better to live under robber barons than under omnipotent moral busybodies. The robber baron's cruelty may sometimes sleep, his cupidity may at some point be satiated; but those who torment us for our own good will torment us without end, for they do so with the approval of their own conscience." – C. S. Lewis
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