Class 8 Maths

# 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

 22 4024 2 4 002

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

(ii) 1989

 44 198916 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

 55 325025 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

 22 8254 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

 66 400036 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

 22 5254 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

 44 175016 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

 11 2521 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

 44 182516 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

 88 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.

 33 10009 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.

 22 5004 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.