Calculate by hand $$\sqrt{471.969}$$

### Development:

We group the digits two by two and $$47.19.69$$ is obtained in the radicand. We look for a number that, when squared, gives $$47$$. The closest number is $$6$$ because $$6\cdot6 = 36$$.

47.19.69 | 6 |

6·6=36 |

This result is subtracted from $$47$$ and the following digits $$(19)$$ are moved down. We also separate the last number.

47.19.69 | 6 |

-36 | 6·6=36 |

111.9 |

Then write the double of $$6$$, which is $$12$$. We divide $$111$$ by $$12$$, to obtain the number that it is necessary to add and to multiply. In this case it is $$8$$.

It is subtracted and the following group of two digits are moved down and the last digit is separated.

47.19.69 | 68 |

-36 | 6·6=36 |

111.9 | 6·2=12 |

-1024 | 128·8=1024 |

956.9 |

We repeat the previous step.

The double of $$68$$ is written below, $$136$$. Divide $$956$$ by $$136$$, to obtain the number that it is necessary to add and to multiply. In this case it is $$7$$, which must be added to the first line.

Operating the remainder is zero, therefore, the first line is taken and that number is the root.

47.19.69 | 687 |

-36 | 6·6=36 |

111.9 | 6·2=12 |

-1024 | 128·8=1024 |

956.9 | 68·2=136 |

-9569 | 1367·7=9569 |

0 |

### Solution:

$$\sqrt{471969}=687$$