# Square Roots

## Exercise 6.4

### Part 2

Question 4: Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(i) 402

2 2 | 402 4 | 2 |

4 | 002 |

It is clear that if 2 is subtracted then we will get 400, which is a perfect square.

(ii) 1989

4 4 | 1989 16 | 4 |

8 | 389 |

Here, `84 xx 4 = 336` which is less than 389

And, `85 xx 5 = 425`, which is more than 389

Hence the required difference `= 389 - 336 = 53`

`1989 - 53 = 1936` is a perfect square.

(iii) 3250

5 5 | 3250 25 | 5 |

10 | 750 |

Here, `107 xx 7 = 749` is less than 750

`108 xx 8 = 864` is more than 750

Hence, the required difference `= 750 - 749 = 1`

`3250 - 1 = 3249` is a perfect square.

(iv) 825

2 2 | 825 4 | 2 |

4 | 425 |

Here, `48 xx 8 = 384` is less than 425

`49 xx 9 = 441` is more than 425

Hence, the required difference `= 425 - 384 = 41`

`825 - 41 = 784` is a perfect square.

(v) 4000

6 6 | 4000 36 | 6 |

12 | 400 |

Here, `123 xx 3 = 369` is less than 400

`124 xx 4 = 496` is more than 400

Hence, the required difference `= 400 - 369 = 31`

`4000 - 31 = 3969` is a perfect square.

Question 5: Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(i) 525

2 2 | 525 4 | 2 |

4 | 125 |

Here, `43 xx 3 = 129` is more than 125

`42 xx 2 = 84` is less than 125

Hence, required addition `= 129 - 125 = 4`

`525 + 4 = 529` is a perfect square.

(ii) 1750

4 4 | 1750 16 | 4 |

8 | 150 |

Here, `81xx1=81` is less 150

And, `82xx2=164` which is 14 more than 150

So, `1750+14=1764` is a perfect square

(iii) 252

1 1 | 252 1 | 1 |

2 | 152 | |

Here, `25 xx 5 = 125` is less than 152

`26 xx 6 = 156` is more than 152

Required difference `= 156 - 152 = 4`

So, `252 + 4 = 256` is a perfect square

(iv) 1825

4 4 | 1825 16 | 4 |

8 | 225 |

Here, `82 xx 2 = 164` is less than 225

`83 xx 3 = 249` is more than 225

Required difference `= 249 - 225 = 24`

So, `1825 + 24 = 1849` is a perfect square

(v) 6412

8 8 | 6412 | 8 |

16 | 12 | |

Here, we need `161 xx 1 = 161`

Required difference `= 161 - 12 = 149`

So, `6412 + 149 = 6561` is a perfect square

Question 6: Find the length of the side of a square whose area is 441 m².

**Answer:** Area of Square = Side²

Side `=sqrt\text(Area)`

`441=3xx3xx7xx7`

Or, `sqrt(441)=3xx7=21`

(a) If AB = 6 cm, BC = 8 cm, find AC

**Answer:** `= AC^2= AB^2 + BC^2`

`= 6^2 + 8^2 = 36 + 64 = 100`

`AC=sqrt(100)=10`

(b) If AC = 13 cm, BC = 5 cm, find AB

**Answer:** `AB^2= AC^2- BC^2`

`= 13^2- 5^2= 169 - 25 = 144`

`AB=sqrt(144)=12`

Question 8: A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.

3 3 | 1000 9 | 3 |

6 | 100 |

Here, `61xx1=61` is less than 100

`62xx2=124` is more than 100

Hence, the required difference `= 100-61=39`

Min. number of plants required `= 1000-39=961`

Question 9: There are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement.

2 2 | 500 4 | 2 |

4 | 100 |

Here, `42 xx 2 = 84` is less than 100

`43 xx 3 = 129` is more than 100

Hence, the required difference `= 100 - 84 = 16`

So, 16 children will be left out in the arrangement.