Square Roots

Exercise 6.4

Question 1: Find the square root of each of the following numbers by Division method.

(i) 2304

(ii) 4489

(iii) 529

(iv) 3249

(v) 1369

(vi) 5776

(vii) 7921

(viii) 576

(ix) 1024

(x) 3136

(xi) 900

Question 2: Find the number of digits in the square root of each of the following numbers (without any calculation).

(i) 64 (ii) 144 (iii) 4489 (iv) 27225 (v) 390625

Answer: If there are even number of digits in square then number of digits in

Square Root =(π)/(2)

If there are odd number of digits in square then number of digits in

Square Root =(π+1)/(2)

(i) 1, (ii) 2, (iii) 2, (iv) 3, (v) 3

3. Find the square root of the following decimal numbers.

(i) 2.56

(ii) 7.29

(iii) 51.84

(iv) 42.25

(v) 31.36

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

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

(ii) 1989

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

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

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

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

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

Here, 161 xx 1 = 161 is 11 more than 150
So, 1750 + 11 = 1761 is a perfect square

(iii) 252

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

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

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.

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.

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.