In this tutorial, we are going to learn the Mathematical shortcut to obtain the square root of any perfect square.

Once you practice this shortcut, no matter how big the number is, you can easily tell the square-root of the number. Please note, this shortcut works only with perfect squares

Before we go into this shortcut, it is very important that we know the squares of the number atleast till 30.

Let’s recall the squares of numbers till 10.

Number | Square |

1 | 1 |

2 | 4 |

3 | 9 |

4 | 16 |

5 | 25 |

6 | 36 |

7 | 49 |

8 | 64 |

9 | 81 |

10 | 100 |

Notice the unit-digits of the perfect square numbers.

- If the perfect square ends with 1, the unit digit of square root will be either 1 or 9.
- If the perfect square ends with 4, the unit digit of square root will be either 2 or 8.
- If the perfect square ends with 9, the unit digit of square root will be either 3 or 7.
- If the perfect square ends with 6, the unit digit of square root will be either 4 or 6.
- If the perfect square ends with 5, the unit digit of square root will be always be 5.
- If the perfect square ends with 0, the unit digit of square root will be always be 0.

Last digit of the square |
Last digit of the square-root |

1 |
1 or 9 |

4 |
2 or 8 |

5 |
3 or 7 |

6 |
4 or 6 |

9 |
5 |

0 |
0 |

** **

**So, a perfect square, will never end with the digits 2,3,7 or 8. **

Let’s try to understand the shortcut through an example.

**Exercise : **compute the square-root of the perfect square of 9801.

**Step 1: ** We need to find out the number of digits in the square-roots. If a number has **n **digits, then its square-root will have **n/2 **digits

9801 has 4 digits. So the square root will have 2 digits. __ __

**Step 2: **Check the bound where the square number lies.

9801 lies between 2 perfect square 8100 and 10000. So, the square-root lies between 91 -99. So, the first digit will always be 9.

**Step 3: **Check the unit digit now. From the above table, it is pretty clear now as the unit digit is 1, the square root will have either 1 or 9. So the square root will be either 91 or 99. We are not sure still.

**Step 4: **Now consider the square 9801. Notice that the number is much closer to 10000 than 8100. So, we need to choose the square root which is closer to 100. Choose the biggest of 91 and 99.

So the square root of 9801 is 99.

**Exercise : **Using this shortcut, can you obtain the square of 65536

**Step 1 : ** 65536 has 5 digits. So the square root will have 3 digits. __ __ __

**Step 2 : **Check the bound where the square number lies. 65536 lies between the bounds 40000 and 90000. So the square-root lies between 200 and 300. So the first digit is 2. So the square root of 65536 is 25__.

**Step 3 : **Go a step further. We know 25^{2} is 625 and 26^{2} is 676. So, the number 65536 lies between 62500 and 65600. So, the square-root lies between 250 and 260. So, the second digit would be 5. So the square root of 65536 is 25__.

**Step 4 : **Check the unit digit now. From the above table, it is pretty clear now as the unit digit is 6, the square root will have either 4 or 6. So the square root will be either 254 or 256. We are not sure still.

As the number 65536 is mid-way between 62500 and 67600, we cannot choose either 254 or 256.

To break this deadlock, just calculate 255^{2} using the Shortcut, how to obtain the square of a number ending with 5.

255^{2} = 25*(25+1)/25 = 65025.

As you can see now, 65536 is more than 65025. So the square-root of this number is obviously more than 255. So, the square-root will be 256.

Using this shortcut, can you obtain the square-roots of the following perfect squares ?

- 15129
- 7225
- 1296
- 5041

Please contact me if some point in the tutorial is not clear. I would be glad to help.

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