# Computing nth roots

We saw in section 1.3.3 that attempting to compute square roots by naively finding a fixed point of `y → x/y` does not converge, and that this can be fixed by average damping. The same method works for finding cube roots as fixed points of the average-damped `y → x/y²` . Unfortunately, the process does not work for fourth roots -- a single average damp is not enough to make a fixed-point search for `y → x/y³` converge. On the other hand, if we average damp twice (i.e., use the average damp of the average damp of `y → x/y³` ) the fixed-point search does converge. Do some experiments to determine how many average damps are required to compute `n` th roots as a fixed-point search based upon repeated average damping of `y → x/yⁿ⁻¹` . Use this to implement a simple procedure for computing `n` th roots using `fixed-point` , `average-damp` , and the `repeated` procedure of exercise 1.43 . Assume that any arithmetic operations you need are available as primitives.

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``````(define (root3 x) ((nth-root 3) x))