#
Computing nth roots
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}

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.

```
(define (root3 x) ((nth-root 3) x))
(define (root4 x) ((nth-root 4) x))
(check-equal? (round (root3 27)) 3.0)
(check-equal? (round (root3 64)) 4.0)
(check-equal? (round (root4 81)) 3.0)
(check-equal? (round (root4 10000)) 10.0)
```