## NCERT Solutions for Class 8 Chapter 6 Squares and Square Roots Maths

Page No: 96**Exercise 6.1**

1. What will be the unit digit of the squares of the following numbers?

(i) 81

(ii) 272

(iii) 799

(iv) 3853

(v) 1234

(vi) 26387

(vii) 52698

(viii) 99880

(ix) 12796

(x) 55555

**Solution**

The unit digit of square of a number having 'a' at its unit place ends with aÃ—a.

(i) The unit digit of the square of a number having digit 1 as unitâ€™s place is 1.

âˆ´ Unit digit of the square of number 81 is equal to 1.

(ii) The unit digit of the square of a number having digit 2 as unitâ€™s place is 4.

âˆ´ Unit digit of the square of number 272 is equal to 4.

(iii) The unit digit of the square of a number having digit 9 as unitâ€™s place is 1.

âˆ´ Unit digit of the square of number 799 is equal to 1.

(iv) The unit digit of the square of a number having digit 3 as unitâ€™s place is 9.

âˆ´ Unit digit of the square of number 3853 is equal to 9.

(v) The unit digit of the square of a number having digit 4 as unitâ€™s place is 6.

âˆ´ Unit digit of the square of number 1234 is equal to 6.

(vi) The unit digit of the square of a number having digit 7 as unitâ€™s place is 9.

âˆ´ Unit digit of the square of number 26387 is equal to 9.

(vii) The unit digit of the square of a number having digit 8 as unitâ€™s place is 4.

âˆ´ Unit digit of the square of number 52698 is equal to 4.

(viii) The unit digit of the square of a number having digit 0 as unitâ€™s place is 01.

âˆ´ Unit digit of the square of number 99880 is equal to 0.

(ix) The unit digit of the square of a number having digit 6 as unitâ€™s place is 6.

âˆ´ Unit digit of the square of number 12796 is equal to 6.

(x) The unit digit of the square of a number having digit 5 as unitâ€™s place is 5.

âˆ´ Unit digit of the square of number 55555 is equal to 5.

2. The following numbers are obviously not perfect squares. Give reason.

(i) 1057

(ii) 23453

(iii) 7928

(iv) 222222

(v) 64000

(vi) 89722

(vii) 222000

(viii) 505050

As we know that natural numbers ending in the digits 0, 2, 3, 7 and 8 are not perfect square.

âˆ´ 1057, 23453, 7928, 222222, 64000, 89722, 222000 and 505050 are not perfect squares.

3. The following numbers are obviously not perfect squares. Give reason.

As we know that the square of an odd number is odd and the square of an even number is even. Therefore,

(i) The square of 431 is an odd number.

(ii) The square of 2826 is an even number.

(iii) The square of 7779 is an odd number.

(iv)The square of 82004 is an even number.

4.The squares of which of the following would be odd numbers?

11

101

1001

100001

10000001

We observe that the square on the number on R.H.S of the equality has an odd number of digits such that the first and last digits both are 1 and middle digit is 2. And the number of zeros between left most digits 1 and the middle digit 2 and right most digit 1 and the middle digit 2 is same as the number of zeros in the given number.

âˆ´ 100001

10000001

5. Observe the following pattern and supply the missing numbers.

11

101

10101

1010101

............

We observe that the square on the number on R.H.S of the equality has an odd number of digits such that the first and last digits both are 1. And, the square is symmetric about the middle digit. If the middle digit is 4, then the number to be squared is 10101 and its square is 102030201.

So, 1010101

101010101

6. Using the given pattern, find the missing numbers.

1

2

3

4

5 + _

6 + 7 + _

Given, 1

i.e 1

âˆ´ 2

âˆ´ 3

âˆ´ 4

âˆ´ 5

â€¢ 6

âˆ´ 6

7. Without adding, find the sum.

(i) 1 + 3 + 5 + 7 + 9

(ii) 1 + 3 + 5 + 7 + 9 + I1 + 13 + 15 + 17 +19

(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

8. (i) Express 49 as the sum of 7 odd numbers.

Since, 49 = 7

âˆ´ 49 = sum of first 7 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13

(ii) Express 121 as the sum of 11 odd numbers.

âˆ´ 121 = sum of first 11 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

9. How many numbers lie between squares of the following numbers?

(i) 12 and 13

(ii) 25 and 26

(iii) 99 and 100

Between n

(i) 12

(ii) 25

(iii) 99

1. Find the square of the following numbers.

(i) 32

(ii) 35

(iii) 86

(iv) 93

(v) 71

(vi) 46

(i) (32)

= (30 +2)

= (30)

= 900 + 4 + 120

= 1024

(ii) (35)

= (30+5 )

= (30)

= 900 + 25 + 300

= 1225

(iii) (86)

= (90 - 4)

= (90)

= 8100 + 16 - 720

= 8116 - 720

= 7396

(iv) (93)

= (90+3 )

= (90)

= 8100 + 9 + 540

= 8649

(v) (71)

= (70+1 )

= (70)

= 4900 + 1 + 140

= 5041

(v) (46)

= (50 -4 )

= (50)

= 2500 + 16 - 400

= 2516 - 400

= 2116

2. Write a Pythagorean triplet whose one member is.

(i) 6

(ii) 14

(iii) 16

(iv) 18

For any natural number m, we know that

2m, m

(i) 2m = 6

â‡’ m = 6/2 = 3

m

m

âˆ´ (6, 8, 10) is a Pythagorean triplet.

(ii) 2m = 14

â‡’ m = 14/2 = 7

m

m

âˆ´ (14, 48, 50) is not a Pythagorean triplet.

(iii) 2m = 16

â‡’ m = 16/2 = 8

m

m

âˆ´ (16, 63, 65) is a Pythagorean triplet.

(iv) 2m = 18

â‡’ m = 18/2 = 9

m

m

âˆ´ (18, 80, 82) is a Pythagorean triplet.

(i) 9801

(ii) 99856

(iii) 998001

(iv) 657666025

(i) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 1 is 1.

âˆ´ Unitâ€™s digit of the square root of number 9801 is equal to 1.

(ii) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 6 is 6.

âˆ´ Unitâ€™s digit of the square root of number 99856 is equal to 6.

(iii) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 1 is 1.

âˆ´ Unitâ€™s digit of the square root of number 998001 is equal to 1.

(iv) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 5 is 5.

âˆ´ Unitâ€™s digit of the square root of number 657666025 is equal to 5.

2. Without doing any calculation, find the numbers which are surely not perfect squares.

(i) 153

(ii) 257

(iii) 408

(iv) 441

As we know that natural numbers ending in the digits 0, 2, 3, 7 and 8 are not perfect square.

âˆ´ 153, 257, 408 are not perfect squares and 441 is a perfect square.

3. Find the square roots of 100 and 169 by the method of repeated subtraction.

â€¢ 100 â€“ 1 = 99

99 â€“ 3 = 96

96 â€“ 5 = 91

91 â€“ 7 = 84

84 â€“ 9 = 75

75 â€“ 11 = 64

64 â€“ 13 = 51

51 â€“ 15 = 36

36 â€“ 17 = 19

19 â€“ 19 = 0

Therefore, we have performed subtraction tenth times.

âˆ´ âˆš100 = 10

â€¢ 169 â€“ 1 = 168

168 â€“ 3 = 165

165 â€“ 5 = 160

160 â€“ 7 = 153

153 â€“ 9 = 144

144 â€“ 11 = 133

133 â€“ 13 = 120

120 â€“ 15 = 105

105 â€“ 17 = 88

88 â€“ 19 = 69

69 â€“ 21 = 48

48 â€“ 23 = 25

25 â€“ 25 = 0

Therefore, we have performed subtraction thirteenth times.

âˆ´ âˆš169 = 13

4.Find the square roots of the following numbers by the Prime Factorisation Method.

(i) 729

(ii) 400

(iii) 1764

(iv) 4096

(v) 7744

(vi) 9604

(vii) 5929

(viii) 9216

(ix) 529

(x) 8100

(i)

(ii)

â‡’ âˆš400 = 2Ã—2Ã—5 = 20âˆ´ Unit digit of the square of number 1234 is equal to 6.

(vi) The unit digit of the square of a number having digit 7 as unitâ€™s place is 9.

âˆ´ Unit digit of the square of number 26387 is equal to 9.

(vii) The unit digit of the square of a number having digit 8 as unitâ€™s place is 4.

âˆ´ Unit digit of the square of number 52698 is equal to 4.

(viii) The unit digit of the square of a number having digit 0 as unitâ€™s place is 01.

âˆ´ Unit digit of the square of number 99880 is equal to 0.

(ix) The unit digit of the square of a number having digit 6 as unitâ€™s place is 6.

âˆ´ Unit digit of the square of number 12796 is equal to 6.

(x) The unit digit of the square of a number having digit 5 as unitâ€™s place is 5.

âˆ´ Unit digit of the square of number 55555 is equal to 5.

2. The following numbers are obviously not perfect squares. Give reason.

(i) 1057

(ii) 23453

(iii) 7928

(iv) 222222

(v) 64000

(vi) 89722

(vii) 222000

(viii) 505050

**Solution**As we know that natural numbers ending in the digits 0, 2, 3, 7 and 8 are not perfect square.

âˆ´ 1057, 23453, 7928, 222222, 64000, 89722, 222000 and 505050 are not perfect squares.

3. The following numbers are obviously not perfect squares. Give reason.

**Solution**As we know that the square of an odd number is odd and the square of an even number is even. Therefore,

(i) The square of 431 is an odd number.

(ii) The square of 2826 is an even number.

(iii) The square of 7779 is an odd number.

(iv)The square of 82004 is an even number.

4.The squares of which of the following would be odd numbers?

11

^{2}= 121101

^{2}= 102011001

^{2}= 1002001100001

^{2}= 1 .......2.........110000001

^{2}= ..........................**Solution**We observe that the square on the number on R.H.S of the equality has an odd number of digits such that the first and last digits both are 1 and middle digit is 2. And the number of zeros between left most digits 1 and the middle digit 2 and right most digit 1 and the middle digit 2 is same as the number of zeros in the given number.

âˆ´ 100001

^{2 }= 1000020000110000001

^{2}^{ }= 1000000200000015. Observe the following pattern and supply the missing numbers.

11

^{2}= 121101

^{2}= 1020110101

^{2}= 1020302011010101

^{2}= .......................................

^{2}= 10203040504030201**Solution**We observe that the square on the number on R.H.S of the equality has an odd number of digits such that the first and last digits both are 1. And, the square is symmetric about the middle digit. If the middle digit is 4, then the number to be squared is 10101 and its square is 102030201.

So, 1010101

^{2 }=1020304030201101010101

^{2}^{ }=102030405050302016. Using the given pattern, find the missing numbers.

1

^{2}+ 2^{2}+ 2^{2}= 3^{2}2

^{2}+ 3^{2}+ 6^{2}= 7^{2}3

^{2}+ 4^{2}+ 12^{2}= 13^{2}4

^{2}+ 5^{2}+ _^{2 }= 21^{2}5 + _

^{2}+ 30^{2}= 31^{2}6 + 7 + _

^{2}= __^{2}**Solution**Given, 1

^{2}^{ }^{ }+ 2^{2}^{ }+ 2^{2 }= 3^{2}i.e 1

^{2 }+ 2^{2 }+ (1Ã—2 )^{2 }= ( 1^{2 }+ 2^{2 }-1 Ã— 2 )^{2}^{}â€¢ 2^{2 }+ 3^{2 }+ 6^{2 }=7^{2}âˆ´ 2

^{2 }+ 3^{2 }+ (2Ã—3 )^{2 }= (2^{2 }+ 3^{2 }-2 Ã— 3)^{2 }^{}â€¢ 3^{2 }+ 4^{2 }+ 12^{2 }= 13^{2}âˆ´ 3

^{2 }+ 4^{2 }+ (3Ã—4 )^{2 }= (3^{2 }+ 4^{2 }- 3 Ã— 4)^{2 }^{}â€¢ 4^{2 }+ 5^{2 }+ (4Ã—5 )^{2 }= (4^{2 }+ 5^{2 }- 4 Ã— 5)^{2 }âˆ´ 4

^{2 }+ 5^{2 }+ 20^{2 }= 21^{2}^{}â€¢ 5^{2 }+ 6^{2 }+ (5Ã—6 )^{2 }= (5^{2}+ 6^{2 }- 5 Ã— 6)^{2}âˆ´ 5

^{2 }+ 6^{2 }+ 30^{2 }= 31^{2}â€¢ 6

^{2 }+ 7^{2 }+ (6Ã—7 )^{2 }= (6^{2 }+ 7^{2 }- 6 Ã— 7)^{2 }âˆ´ 6

^{2 }+ 7^{2}+ 42^{2 }= 43^{2}7. Without adding, find the sum.

(i) 1 + 3 + 5 + 7 + 9

**Solution****Sum of first five odd number = (5)**

^{2 }= 25(ii) 1 + 3 + 5 + 7 + 9 + I1 + 13 + 15 + 17 +19

**Solution****Sum of first ten odd number = (10)**

^{2 }= 100(iii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

**Solution****Sum of first thirteen odd number = (12)**

^{2 }= 1448. (i) Express 49 as the sum of 7 odd numbers.

**Solution****We know, sum of first n odd natural numbers is n**

^{2 }.Since, 49 = 7

^{2 }âˆ´ 49 = sum of first 7 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13

(ii) Express 121 as the sum of 11 odd numbers.

**Solution****Since, 121 = 11**

^{2}âˆ´ 121 = sum of first 11 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

9. How many numbers lie between squares of the following numbers?

(i) 12 and 13

(ii) 25 and 26

(iii) 99 and 100

**Solution**Between n

^{2}and (n+1)^{2},^{ }there are 2n nonâ€“perfect square numbers.(i) 12

^{2 }and 13^{2}there are 2Ã—12 = 24 natural numbers.(ii) 25

^{2}and 26^{2}there are 2Ã—25 = 50 natural numbers.(iii) 99

^{2}and 100^{2 }there are 2Ã—99 =198 natural numbers.**Exercise 6.2**1. Find the square of the following numbers.

(i) 32

(ii) 35

(iii) 86

(iv) 93

(v) 71

(vi) 46

**Solution**(i) (32)

^{2}= (30 +2)

^{2}= (30)

^{2 }+ (2)^{2 }+ 2Ã—30Ã—2 [Since, (a+b)^{2 }= a^{2}+b^{2 }+2ab]= 900 + 4 + 120

= 1024

(ii) (35)

^{2}= (30+5 )

^{2}= (30)

^{2 }+ (5)^{2 }+ 2Ã—30Ã—5 [Since, (a+b)^{2 }= a^{2}+b^{2 }+2ab]= 900 + 25 + 300

= 1225

(iii) (86)

^{2}= (90 - 4)

^{2}= (90)

^{2 }+ (4)^{2 }- 2Ã—90Ã—4 [Since, (a-b)^{2 }= a^{2 }+ b^{2 }- 2ab]= 8100 + 16 - 720

= 8116 - 720

= 7396

(iv) (93)

^{2}= (90+3 )

^{2}= (90)

^{2 }+ (3)^{2 }+ 2Ã—90Ã—3 [Since, (a+b)^{2 }= a^{2}+b^{2 }+2ab]= 8100 + 9 + 540

= 8649

(v) (71)

^{2}= (70+1 )

^{2}= (70)

^{2 }+ (1)^{2 }+2Ã—70Ã—1 [Since, (a+b)^{2 }= a^{2}+b^{2 }+2ab]= 4900 + 1 + 140

= 5041

(v) (46)

^{2}= (50 -4 )

^{2}= (50)

^{2 }+ (4)^{2 }- 2Ã—50Ã—4 [Since, (a-b)^{2 }= a^{2 }+ b^{2 }- 2ab]= 2500 + 16 - 400

= 2516 - 400

= 2116

2. Write a Pythagorean triplet whose one member is.

(i) 6

(ii) 14

(iii) 16

(iv) 18

**Solution**For any natural number m, we know that

2m, m

^{2}â€“1, m^{2}+1 is a Pythagorean triplet.(i) 2m = 6

â‡’ m = 6/2 = 3

m

^{2}â€“1= 3^{2}â€“ 1 = 9â€“1 = 8m

^{2}+1= 3^{2}+1 = 9+1 = 10âˆ´ (6, 8, 10) is a Pythagorean triplet.

(ii) 2m = 14

â‡’ m = 14/2 = 7

m

^{2}â€“1= 7^{2}â€“1 = 49â€“1 = 48m

^{2}+1 = 7^{2}+1 = 49+1 = 50âˆ´ (14, 48, 50) is not a Pythagorean triplet.

(iii) 2m = 16

â‡’ m = 16/2 = 8

m

^{2}â€“1 = 8^{2}â€“1 = 64â€“1 = 63m

^{2}+ 1 = 8^{2}+1 = 64+1 = 65âˆ´ (16, 63, 65) is a Pythagorean triplet.

(iv) 2m = 18

â‡’ m = 18/2 = 9

m

^{2}â€“1 = 9^{2}â€“1 = 81â€“1 = 80m

^{2}+1 = 9^{2}+1 = 81+1 = 82âˆ´ (18, 80, 82) is a Pythagorean triplet.

**Exercise 6.3****1.What could be the possible â€˜oneâ€™sâ€™ digits of the square root of each of the following numbers?**

(i) 9801

(ii) 99856

(iii) 998001

(iv) 657666025

**Solution**(i) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 1 is 1.

âˆ´ Unitâ€™s digit of the square root of number 9801 is equal to 1.

(ii) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 6 is 6.

âˆ´ Unitâ€™s digit of the square root of number 99856 is equal to 6.

(iii) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 1 is 1.

âˆ´ Unitâ€™s digit of the square root of number 998001 is equal to 1.

(iv) As we know that the unitâ€™s digit of the square of a number having digit as unitâ€™s place 5 is 5.

âˆ´ Unitâ€™s digit of the square root of number 657666025 is equal to 5.

2. Without doing any calculation, find the numbers which are surely not perfect squares.

(i) 153

(ii) 257

(iii) 408

(iv) 441

**Solution**As we know that natural numbers ending in the digits 0, 2, 3, 7 and 8 are not perfect square.

âˆ´ 153, 257, 408 are not perfect squares and 441 is a perfect square.

3. Find the square roots of 100 and 169 by the method of repeated subtraction.

**Solution**â€¢ 100 â€“ 1 = 99

99 â€“ 3 = 96

96 â€“ 5 = 91

91 â€“ 7 = 84

84 â€“ 9 = 75

75 â€“ 11 = 64

64 â€“ 13 = 51

51 â€“ 15 = 36

36 â€“ 17 = 19

19 â€“ 19 = 0

Therefore, we have performed subtraction tenth times.

âˆ´ âˆš100 = 10

â€¢ 169 â€“ 1 = 168

168 â€“ 3 = 165

165 â€“ 5 = 160

160 â€“ 7 = 153

153 â€“ 9 = 144

144 â€“ 11 = 133

133 â€“ 13 = 120

120 â€“ 15 = 105

105 â€“ 17 = 88

88 â€“ 19 = 69

69 â€“ 21 = 48

48 â€“ 23 = 25

25 â€“ 25 = 0

Therefore, we have performed subtraction thirteenth times.

âˆ´ âˆš169 = 13

4.Find the square roots of the following numbers by the Prime Factorisation Method.

(i) 729

(ii) 400

(iii) 1764

(iv) 4096

(v) 7744

(vi) 9604

(vii) 5929

(viii) 9216

(ix) 529

(x) 8100

**Solution**(i)

729 = 3Ã—3Ã—3Ã—3Ã—3Ã—3Ã—1

â‡’ 729 = (3Ã—3)Ã—(3Ã—3)Ã—(3Ã—3)

â‡’ 729 = (3Ã—3Ã—3)Ã—(3Ã—3Ã—3)

â‡’ 729 = (3Ã—3Ã—3)

â‡’ âˆš729 = 3Ã—3Ã—3 = 27^{2}(ii)

400 = 2Ã—2Ã—2Ã—2Ã—5Ã—5Ã—1

â‡’ 400 = (2Ã—2)Ã—(2Ã—2)Ã—(5Ã—5)

â‡’ 400 = (2Ã—2Ã—5)Ã—(2Ã—2Ã—5)

â‡’ 400 = (2Ã—2Ã—5)

^{2}
1764 = 2Ã—2Ã—3Ã—3Ã—7Ã—7

â‡’ 1764 = (2Ã—2)Ã—(3Ã—3)Ã—(7Ã—7)

â‡’ 1764 = (2Ã—3Ã—7)Ã—(2Ã—3Ã—7)

â‡’ 1764 = (2Ã—3Ã—7)

^{2}
â‡’ âˆš1764 = 2 Ã—3Ã—7 = 42

(iv)

4096 = 2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2

â‡’ 4096 = (2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)

â‡’ 4096 = (2Ã—2Ã—2Ã—2Ã—2Ã—2)Ã—(2Ã—2Ã—2Ã—2Ã—2Ã—2)

â‡’ 4096 = (2Ã—2Ã—2Ã—2Ã—2Ã—2)

^{2}
â‡’ âˆš4096 = 2Ã—2Ã—2 Ã—2Ã—2Ã—2 = 64

7744 = 2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—11Ã—11Ã—1

â‡’ 7744 = (2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(11Ã—11)

â‡’ 7744 = (2Ã—2Ã—2Ã—11)Ã—(2Ã—2Ã—2Ã—11)

â‡’ 7744 = (2Ã—2Ã—2Ã—11)

^{2}
â‡’ âˆš7744 = 2Ã—2Ã—2Ã—11 = 88

(vi)

9604 = 62 Ã— 2 Ã— 7 Ã— 7 Ã— 7 Ã— 7

â‡’ 9604 = ( 2 Ã— 2 ) Ã— ( 7 Ã— 7 ) Ã— ( 7 Ã— 7 )

â‡’ 9604 = ( 2 Ã— 7 Ã—7 ) Ã— ( 2 Ã— 7 Ã—7 )

â‡’ 9604 = ( 2Ã—7Ã—7 )

^{2}
â‡’ âˆš9604 = 2Ã—7Ã—7

= 98

(vii)

5929 = 7Ã—7Ã—11Ã—11

â‡’ 5929 = (7Ã—7)Ã—(11Ã—11)

â‡’ 5929 = (7Ã—11)Ã—(7Ã—11)

â‡’ 5929 = (7Ã—11)

^{2}
â‡’ âˆš5929 = 7Ã—11 = 77

(viii)

(i)

(i)

(ii)

âˆ´ âˆš4489 = 67

(iii)

âˆ´ âˆš3481 = 59

(iv)

âˆ´ âˆš529 = 29

(v)

âˆ´ âˆš3249 = 57

(vi)

âˆ´ âˆš1369 = 37

(vii)

âˆ´ âˆš5776 = 76

(viii)

âˆ´ âˆš7921 = 89

(ix)

âˆ´ âˆš576 = 24

(x)

9216 = 2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—3Ã—3Ã—1

â‡’ 9216 = (2Ã—2)Ã—(2Ã—2) Ã— ( 2 Ã— 2 ) Ã— ( 2 Ã— 2 ) Ã— ( 2 Ã— 2 ) Ã— ( 3 Ã— 3 )

â‡’ 9216 = ( 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 3) Ã— ( 2 Ã— 2 Ã— 2 Ã— 2 Ã— 2 Ã— 3)

â‡’ 9216 = 96 Ã— 96

â‡’ 9216 = ( 96 )

^{2}
â‡’ âˆš9216 = 96

(ix)

529 = 23Ã—23

â‡’ 529 = (23)

^{2}
â‡’ âˆš529 = 23

(x)

8100 = 2Ã—2Ã—3Ã—3Ã—3Ã—3Ã—5Ã—5Ã—1

â‡’ 8100 = (2Ã—2) Ã—(3Ã—3)Ã—(3Ã—3)Ã—(5Ã—5)

â‡’ 8100 = (2Ã—3Ã—3Ã—5)Ã—(2Ã—3Ã—3Ã—5)

â‡’ 8100 = 90Ã—90

â‡’ 8100 = (90)

^{2}
â‡’ âˆš8100 = 90

5.For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.

(i) 252

(ii) 180

(iii) 1008

(iv) 2028

(v) 1458

(vi) 768

**Solution**

(i)

252 = 2Ã—2Ã—3Ã—3Ã—7

= (2Ã—2)Ã—(3Ã—3)Ã—7

Here, 7 cannot be paired.

âˆ´ We will multiply 252 by 7 to get perfect square.

New number = 252 Ã— 7 = 1764

1764 = 2Ã—2Ã—3Ã—3Ã—7Ã—7

â‡’ 1764 = (2Ã—2)Ã—(3Ã—3)Ã—(7Ã—7)

â‡’ 1764 = 2

^{2}Ã—3^{2}Ã—7^{2}
â‡’ 1764 = (2Ã—3Ã—7)

^{2}
â‡’ âˆš1764 = 2Ã—3Ã—7 = 42

(ii)

180 = 2Ã—2Ã—3Ã—3Ã—5

= (2Ã—2)Ã—(3Ã—3)Ã—5

Here, 5 cannot be paired.

âˆ´ We will multiply 180 by 5 to get perfect square.

New number = 180 Ã— 5 = 900

900 = 2Ã—2Ã—3Ã—3Ã—5Ã—5Ã—1

â‡’ 900 = (2Ã—2)Ã—(3Ã—3)Ã—(5Ã—5)

â‡’ 900 = 2

^{2}Ã—3^{2}Ã—5^{2}
â‡’ 900 = (2Ã—3Ã—5)

^{2}
â‡’ âˆš900 = 2Ã—3Ã—5 = 30

(iii)

1008 = 2Ã—2Ã—2Ã—2Ã—3Ã—3Ã—7

= (2Ã—2)Ã—(2Ã—2)Ã—(3Ã—3)Ã—7

Here, 7 cannot be paired.

âˆ´ We will multiply 1008 by 7 to get perfect square.

New number = 1008Ã—7 = 7056

7056 = 2Ã—2Ã—2Ã—2Ã—3Ã—3Ã—7Ã—7

â‡’ 7056 = (2Ã—2)Ã—(2Ã—2)Ã—(3Ã—3)Ã—(7Ã—7)

â‡’ 7056 = 2

^{2}Ã—2^{2}Ã—3^{2}Ã—7^{2}
â‡’ 7056 = (2Ã—2Ã—3Ã—7)

^{2}
â‡’ âˆš7056 = 2Ã—2Ã—3Ã—7 = 84

(iv)

2028 = 2Ã—2Ã—3Ã—13Ã—13

= (2Ã—2)Ã—(13Ã—13)Ã—3

Here, 3 cannot be paired.

âˆ´ We will multiply 2028 by 3 to get perfect square.

New number = 2028Ã—3 = 6084

6084 = 2Ã—2Ã—3Ã—3Ã—13Ã—13

â‡’ 6084 = (2Ã—2)Ã—(3Ã—3)Ã—(13Ã—13)

â‡’ 6084 = 2

^{2}Ã—3^{2}Ã—13^{2}
â‡’ 6084 = (2Ã—3Ã—13)

^{2}
â‡’ âˆš6084 = 2Ã—3Ã—13 = 78

(v)

1458 = 2Ã—3Ã—3Ã—3Ã—3Ã—3Ã—3

= (3Ã—3)Ã—(3Ã—3)Ã—(3Ã—3)Ã—2

Here, 3 cannot be paired.

âˆ´ We will multiply 1458 by 2 to get perfect square.

New number = 1458 Ã— 2 = 2916

2916 = 2Ã—2Ã—3Ã—3Ã—3Ã—3Ã—3Ã—3

â‡’ 2916 = (3Ã—3)Ã—(3Ã—3)Ã—(3Ã—3)Ã—(2Ã—2)

â‡’ 2916 = 3

^{2}Ã—3^{2}Ã—3^{2}Ã—2^{2}
â‡’ 2916 = (3Ã—3Ã—3Ã—2)

^{2}
â‡’ âˆš2916 = 3Ã—3Ã—3Ã—2 = 54

(vi)

768 = 2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—3

= (2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—3

Here, 3 cannot be paired.

âˆ´ We will multiply 768 by 3 to get perfect square.

New number = 768Ã—3 = 2304

2304 = 2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—3Ã—3

â‡’ 2304 = (2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(2Ã—2)Ã—(3Ã—3)

â‡’ 2304 = 2

^{2}Ã—2^{2}Ã—2^{2}Ã—2^{2}Ã—3^{2}
â‡’ 2304 = (2Ã—2Ã—2Ã—2Ã—3)

^{2}
â‡’ âˆš2304 = 2Ã—2Ã—2Ã—2Ã—3 = 48

6. For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.

(i) 252

(ii) 2925

(iii) 396

(iv) 2645

(v) 2800

(vi) 1620

(ii) 2925

(iii) 396

(iv) 2645

(v) 2800

(vi) 1620

**Solution**

(i)

252 = 2Ã—2Ã—3Ã—3Ã—7

= (2Ã—2)Ã—(3Ã—3)Ã—7

Here, 7 cannot be paired.

âˆ´ We will divide 252 by 7 to get perfect square.

New number = 252 Ã· 7 = 36

36 = 2Ã—2Ã—3Ã—3

â‡’ 36 = (2Ã—2)Ã—(3Ã—3)

â‡’ 36 = 2

^{2}Ã—3^{2}
â‡’ 36 = (2Ã—3)

^{2}
â‡’ âˆš36 = 2Ã—3 = 6

(ii)

2925 = 3Ã—3Ã—5Ã—5Ã—13

= (3Ã—3)Ã—(5Ã—5)Ã—13

Here, 13 cannot be paired.

âˆ´ We will divide 2925 by 13 to get perfect square.

New number = 2925 Ã· 13 = 225

225 = 3Ã—3Ã—5Ã—5

â‡’ 225 = (3Ã—3)Ã—(5Ã—5)

â‡’ 225 = 3

^{2}Ã—5^{2}
â‡’ 225 = (3Ã—5)

^{2}
â‡’ âˆš36 = 3Ã—5 = 15

(iii)

396 = 2Ã—2Ã—3Ã—3Ã—11

= (2Ã—2)Ã—(3Ã—3)Ã—11

Here, 11 cannot be paired.

âˆ´ We will divide 396 by 11 to get perfect square.

New number = 396 Ã· 11 = 36

36 = 2Ã—2Ã—3Ã—3

â‡’ 36 = (2Ã—2)Ã—(3Ã—3)

â‡’ 36 = 2

^{2}Ã—3^{2}
â‡’ 36 = (2Ã—3)

^{2}
â‡’ âˆš36 = 2Ã—3 = 6

(iv)

2645 = 5Ã—23Ã—23

â‡’ 2645 = (23Ã—23)Ã—5

Here, 5 cannot be paired.

âˆ´ We will divide 2645 by 5 to get perfect square.

New number = 2645 Ã· 5 = 529

529 = 23Ã—23

â‡’ 529 = (23)

^{2}
â‡’ âˆš529 = 23

(v)

2800 = 2Ã—2Ã—2Ã—2Ã—5Ã—5Ã—7

= (2Ã—2)Ã—(2Ã—2)Ã—(5Ã—5)Ã—7

Here, 7 cannot be paired.

âˆ´ We will divide 2800 by 7 to get perfect square.

New number = 2800 Ã· 7 = 400

400 = 2Ã—2Ã—2Ã—2Ã—5Ã—5

â‡’ 400 = (2Ã—2)Ã—(2Ã—2)Ã—(5Ã—5)

â‡’ 400 = (2Ã—2Ã—5)

^{2}
â‡’ âˆš400 = 20

(vi)

1620 = 2Ã—2Ã—3Ã—3Ã—3Ã—3Ã—5

= (2Ã—2)Ã—(3Ã—3)Ã—(3Ã—3)Ã—5

Here, 5 cannot be paired.

âˆ´ We will divide 1620 by 5 to get perfect square.

New number = 1620 Ã· 5 = 324

324 = 2Ã—2Ã—3Ã—3Ã—3Ã—3

â‡’ 324 = (2Ã—2)Ã—(3Ã—3)Ã—(3Ã—3)

â‡’ 324 = (2Ã—3Ã—3)

^{2}
â‡’ âˆš324 = 18

7. The students of Class VIII of a school donated Rs 2401 in all, for Prime Ministerâ€™s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.

**Solution**

Let the number of students in the school be, x.

âˆ´ Each student donate Rs.x .

Total many contributed by all the students = xÃ—x = x

^{2}
Given, x

^{2}= Rs.2401
x

^{2}= 7Ã—7Ã—7Ã—7
â‡’ x

^{2}= (7Ã—7)Ã—(7Ã—7)
â‡’ x

^{2}= 49Ã—49
â‡’ x =

â‡’ x = 49

âˆ´ The number of students = 49

8. 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

**Solution**

Let the number of rows be, x.

âˆ´ the number of plants in each rows = x.

Total many contributed by all the students = xÃ—x = x

^{2}
Given,

x

^{2}= Rs.2025
x

^{2}= 3Ã—3Ã—3Ã—3Ã—5Ã—5
â‡’ x

^{2}= (3Ã—3)Ã—(3Ã—3)Ã—(5Ã—5)
â‡’ x

^{2}= (3Ã—3Ã—5)Ã—(3Ã—3Ã—5)
â‡’ x

^{2}= 45Ã—45
â‡’ x =

â‡’ x = 45

âˆ´ The number of rows = 45 and the number of plants in each rows = 45.

9. Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.

**Solution**

L.C.M of 4, 9 and 10 is (2Ã—2Ã—9Ã—5) 180.

180 = 2Ã—2Ã—9Ã—5

= (2Ã—2)Ã—3Ã—3Ã—5

= (2Ã—2)Ã—(3Ã—3)Ã—5

Here, 5 cannot be paired.

âˆ´ we will multiply 180 by 5 to get perfect square.

Hence, the smallest square number divisible by 4, 9 and 10 = 180Ã—5 = 900

10. Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.

**Solution**

L.C.M of 8, 15 and 20 is (2Ã—2Ã—5Ã—2Ã—3) 120.

120 = 2Ã—2Ã—3Ã—5Ã—2

= (2Ã—2)Ã—3Ã—5Ã—2

Here, 3, 5 and 2 cannot be paired.

âˆ´ We will multiply 120 by (3Ã—5Ã—2) 30 to get perfect square.

Hence, the smallest square number divisible by 8, 15 and 20 =120Ã—30 = 3600

**Exercise 6.4**
1.Find the square root of each of the following numbers by Division method.

(i) 2304

(ii) 4489

(iii) 3481

(iv) 529

(v) 3249

(vi) 1369

(vii) 5776

(viii) 7921

(ix) 576

(x) 1024

(xi) 3136

(xii) 900

**Solution**

(i)

âˆ´ âˆš2304 = 48

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

âˆ´ âˆš1024 = 32

(xi)

âˆ´ âˆš3136 = 56

(xii)

âˆ´ âˆš900 = 30

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

**Solution**

(i)

âˆ´ âˆš64 = 8

Hence, the square root of the number 64 has 1 digit.

(ii)

âˆ´ âˆš144 = 12

Hence, the square root of the number 144 has 2 digits.

(iii)

âˆ´ âˆš4489 = 67

Hence, the square root of the number 4489 has 2 digits.

(iv)

âˆ´ âˆš27225 = 165

Hence, the square root of the number 27225 has 3 digits.

(v)

âˆ´ âˆš390625 = 625

Hence, the square root of the number 390625 has 3 digits.

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

**Solution**

(i)

âˆ´ âˆš2.56 = 1.6

(ii)

âˆ´ âˆš7.29 = 2.7

(iii)

âˆ´ âˆš51.84 = 7.2

(iv)

âˆ´ âˆš42.25 = 6.5

(v)

âˆ´ âˆš31.36 = 5.6

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

(ii) 1989

(iii) 3250

(iv) 825

(v) 4000

**Solution**

(i)

âˆ´ We must subtracted 2 from 402 to get a perfect square.

New number = 402 â€“ 2 = 400

âˆ´ âˆš400 = 20

(ii)

âˆ´ We must subtracted 53 from 1989 to get a perfect square.

New number = 1989 â€“ 53 = 1936

âˆ´ âˆš1936 = 44

(iii)

âˆ´ We must subtracted 1 from 3250 to get a perfect square.

New number = 3250 â€“ 1 = 3249

âˆ´ âˆš3249 = 57

(iv)

âˆ´ We must subtracted 41 from 825 to get a perfect square.

New number = 825 â€“ 41 = 784

âˆ´ âˆš784 = 28

(v)

âˆ´ We must subtracted 31 from 4000 to get a perfect square.

New number = 4000 â€“ 31 = 3961

âˆ´ âˆš3136 = 56

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

(ii) 1750

(iii) 252

(iv) 1825

(v) 6412

**Solution**

(i)

Here, (22)

^{2 }< 525 > (23)^{2}
We can say 525 is ( 129 â€“ 125 ) 4 less than (23)

^{2}.
âˆ´ If we add 4 to 525, it will be perfect square.

New number = 525 + 4 = 529

âˆ´ âˆš529 = 23

(ii)

Here, (41)

^{2}< 1750 > (42)^{2}
We can say 1750 is ( 164 â€“ 150 ) 14 less than (42)

^{2}.
âˆ´ If we add 14 to 1750, it will be perfect square.

New number = 1750 + 14 = 1764

âˆ´âˆš1764 = 42

(iii)

Here, (15)

^{2 }< 252 > (16)^{2}
We can say 252 is ( 156 â€“ 152 ) 4 less than (16)

^{2}.
âˆ´ If we add 4 to 252, it will be perfect square.

New number = 252 + 4 = 256

âˆ´ âˆš256 = 16

(iv)

Here, (42)

^{2 }< 1825 > (43)^{2}
We can say 1825 is ( 249 â€“ 225 ) 24 less than (43)

^{2}.
âˆ´ If we add 24 to 1825, it will be perfect square.

New number = 1825 + 24 = 1849

âˆ´ âˆš1849 = 43

(v)

Here, (80)

^{2 }< 6412 > (81)^{2}
We can say 6412 is ( 161 â€“ 12 ) 149 less than (81)

^{2}.
âˆ´ If we add 149 to 6412, it will be perfect square.

New number = 6412 + 149 = 6561

âˆ´ âˆš6561 = 81

6. Find the length of the side of a square whose area is 441 m

^{2}.**Solution**

Let the length of each side of the field = a

Then, area of the field = 441 m

^{2}
â‡’ a

^{2}= 441 m^{2}
â‡’a = âˆš441 m

âˆ´ The length of each side of the field = a m = 21 m.

7. In a right triangle ABC, âˆ B = 90Â°.

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

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

**Solution**

(a)

Given, AB = 6 cm,

BC = 8 cm

Let AC be x cm.

âˆ´ AC

^{2}= AB^{2}+ BC^{2}
Hence, AC = 10 cm.

(b)

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.

Let the number of rows and column be, x.

âˆ´ Total number of row and column= xÃ—x = x

As per question, x

â‡’ x = âˆš1000

Here, (31)

We can say 1000 is ( 124 â€“ 100 ) 24 less than (32)

âˆ´ 24 more plants is needed.

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.

(b)

Given, AC = 13 cm,

BC = 5 cm

Let AB be x cm.

âˆ´ AC

^{2 }= AB^{2 }+ BC^{2}
â‡’ AC

^{2 }- BC^{2 }= AB^{2}
Hence, AB = 12 cm

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.

**Solution**Let the number of rows and column be, x.

âˆ´ Total number of row and column= xÃ—x = x

^{2}As per question, x

^{2}= 1000â‡’ x = âˆš1000

Here, (31)

^{2}< 1000 > (32)^{2}We can say 1000 is ( 124 â€“ 100 ) 24 less than (32)

^{2}.âˆ´ 24 more plants is needed.

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.

**Solution**
Let the number of rows and column be, x.

âˆ´ Total number of row and column= x Ã— x = x

As per question, x

x = âˆš500

âˆ´ Total number of row and column= x Ã— x = x

^{2}As per question, x

^{2}= 500x = âˆš500

Hence, 16 children would be left out in the arrangement.