*Calculating square roots is rather more difficult than either cube roots or fifth roots - you might want to learn these two methods first.*

Following a few simple steps you will be able to instantly calculate the **square root** of the spectator's number.

To perform this, you should ask the spectator to choose **any whole number less than 100** and, using a calculator, to square it by multiplying the number by itself.

The spectator then calls out the answer, and you instantly reveal the original number (i.e., the square root).

**Note that this method will NOT work with non-whole numbers, or with numbers greater than 99**.

To master the system you must **learn by heart** the squares of numbers 0 to 9, which are shown in the table below. You also need to consider the last digit of each square.

Number | Square | Last Digit |
---|---|---|

0 | 0 | 0 |

1 | 1 | 1 |

2 | 4 | 4 |

3 | 9 | 9 |

4 | 16 | 6 |

5 | 25 | 5 |

6 | 36 | 6 |

7 | 49 | 9 |

8 | 64 | 4 |

9 | 81 | 1 |

Note how the last digits for the squares of 1 and 9 are both 1.

Note how the last digits for the squares of 2 and 8 are both 4.

Note how the last digits for the squares of 3 and 7 are both 9.

Note how the last digits for the squares of 4 and 6 are both 6.

**Ignore the last TWO** digits of the number called out by the spectator and choose the memorized **comparison square** which is **just lower (or equal)** to the remaining number. The corresponding square root is the **first digit**. of your answer

Spectator calls 676

- Ignore 76, leaving 6
- The comparison square which is
**just lower**than 6 is 4 - Therefore the first digit of the answer is the square root of 4 = 2

Spectator calls 5184

- Ignore 84, leaving 51
- The comparison square which is
**just lower**than 51 is 49 - Therefore the first digit of the answer is the square root of 49 = 7

Now consider the **last digit** of the number called out by the spectator and compare this to the last digit of the memorized squares.

- If the last digit of the spectator's call is 0, then the last digit of your answer is also 0.
- If the last digit of the spectator's call is 5, then the last digit of your answer is 5.
- In all other cases, the last digit of the spectator's call will indicate
**two possible values**for the last digit of the square root. For example, if the last digit of the spectator's call is 9, then the square root may end in either 3 or 7.

To determine whether the lower or higher of two possible values should be taken, multiply the **first digit** of your answer (as found in Step 2) by **one greater than itself**.

- If this is
**greater than**(i.e., ignoring the last two digits), then the last digit of your answer is the**lower**of the two possible values. - If this is
**less than (or equal to)**the first part of the number called by the spectator, then the last digit of your answer is the**higher**value.

Spectator calls 676

- Last digit of spectator's call = 6
- Therefore the two possible values of the second digit are 4 or 6
- First digit of answer (from Step 2) is 2
- Multiply 2 by (2+1) = 6
- Because 6 is
**equal to**the first part of the spectator's call (6), then the second digit will be the**higher**of the two possible values - Therefore the second digit is 6 (not 4)
- Therefore the square root of 676 is 26

Spectator calls 5184

- Last digit of spectator's call = 4
- Therefore the two possible values of the second digit are 2 or 8
- First digit of answer (from Step 2) is 7
- Multiply 7 by (7+1) = 56
- Because 56 is
**greater than**the first part of the spectator's call (51), then the second digit will be the**lower**of the two possible values - Therefore the second digit is 2 (not 8)
- Therefore the square root of 5184 is 72

Call SQ | 1st Part (P) |
Lower (or =) SQ | 1st Digit (F) | 2nd Digit? | F x (F+1) | Compare with P | 2nd Digit (S) | SQRT = FS |
---|---|---|---|---|---|---|---|---|

144 | 1 | 1 | 1 | 2 or 8 | 2 | 2 > 1 | 2 | 12 |

784 | 7 | 4 | 2 | 2 or 8 | 6 | 6 < 7 | 8 | 28 |

1369 | 13 | 9 | 3 | 3 or 7 | 12 | 12 < 13 | 7 | 37 |

2025 | 20 | 16 | 4 | 5 | 5 | 45 | ||

2401 | 24 | 16 | 4 | 1 or 9 | 20 | 20 < 24 | 9 | 49 |

2809 | 28 | 25 | 5 | 3 or 7 | 30 | 30 > 28 | 3 | 53 |

3600 | 36 | 36 | 6 | 0 | 0 | 60 | ||

5041 | 50 | 49 | 7 | 1 or 9 | 56 | 56 > 50 | 1 | 71 |

7396 | 73 | 64 | 8 | 4 or 6 | 72 | 72 < 73 | 6 | 86 |

8836 | 88 | 81 | 9 | 4 or 6 | 90 | 90 > 88 | 4 | 94 |

Before you try this out on your friends, you should practice until you can calculate the square root instantly and without error.

If you can't do this, then **practice some more**!

To practice and assess your ability, you can use the Square Root Tester shown below.

## Square Root Tester |
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