Exercise 1.45.  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 nth 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 nth 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 (nth-root n x dampings)
  (fixed-point ((repeated average-damp dampings) (lambda (y) (/ x (expt y (- n 1))))) 1.0))

Experimentation suggests that floor(log₂ n) dampings are necessary and 
sufficient for convergence.

(define (nth-root n x)
  (let ((dampings (floor (/ (log n) (log 2)))))
    (fixed-point ((repeated average-damp dampings) (lambda (y) (/ x (expt y (- n 1)))))
                 1.0)))