How do you arrive at a square root?
ne way is to iterate:
Make a guess, square it, compare it with the value you're taking
the root of, and refine your guess based on this comparison.
Repeat until you're as close as you need to be, or until your
guess-refining procedure fails to improve the guess.
A more general way is to take advantage of a property of the
a ln(x) = ln(x^a)
x^a = e^(a ln(x))
The advantage of this approach is that you can use it to do all
kinds of roots and raise numbers to non-integral powers. This
is how slide rules took roots.
Of course, you have to be able to do logarithms.
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Update: June 2012