# Exercise 1.45

We experiment to determine the minimal number of average damps required to compute $$n$$th roots using a fixed-point search.

Here are the result summarized in a table:

$$n$$ minimal number of average damps needed
2 1
3 1
4 2
5 2
6 2
7 2
8 3
9 3
10 3
11 3
12 3
13 3
14 3
15 3
16 4
17 4
31 4
32 5
63 5
64 6

It seems like we need $$\lfloor \log_2 (n) \rfloor$$ average damps to compute $$n$$th roots.

Knowing this information we can write the procedure to approximate $$n$$th roots using the floor, log and expt scheme primitives.

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