# 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.