# 1.8

## Exercise 1.8: Newton’s method for cube roots is based on the fact that if y is an approximation to the cube root of x, then a better approximation is given by the value

$\frac{x/y^2+2y}{3}$

## Use this formula to implement a cube-root procedure analogous to the square- root procedure. (In section Section 1.3.4 [1-3-4], page 69 we will see how to implement Newton’s method in general as an abstraction of these square-root and cube-root procedures.)

```
(define (cube-root lastGuess guess x)
(if (good-enough? lastGuess guess)
guess
(cube-root guess (improve-cube-root guess x) x)
)
)
(define (improve-cube-root guess x)
(/
(+
(/ x (* guess guess))
(* 2 guess)
)
)
3
)
(define (good-enough? lastGuess guess)
(< (abs (- lastGuess guess)) 0.001)
)
```

```
(cube-root 100 1 27)
```