Answer

answer.clj

(defn abs [x]
  "Returns x if the number is > 0, otherwise -x"
  (if (neg? x) (- x) x))

(defn square [x]
  "Squares the number"
  (* x x))

(defn cube [x]
  "Multiplies the square of the number by itself"
  (* (square x) x))

(defn good-enough? [guess x]
  "Returns true if the cube of the guess is less than
   the predefined tolerance (0.001)"
  (<
   (abs (- (cube guess) x))
   0.001))

(defn improve [guess x]
  "Applies the formula of newton's method for cube roots
   to the guess and number"
  (/
   (+
    (/ x (square guess))
    (* 2 guess))
   3))

(defn cbrt-iter [guess x]
  "Recursively apply newton's method until the result is acceptable"
  (if (good-enough? guess x)
    guess
    (cbrt-iter (improve guess x) x)))

(defn cbrt [x]
  "Returns the cube root of a number via newton's method
   with a default guess of 1"
  (cbrt-iter 1.0 x))