NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (2022)

NCERT Solutions Class 8 Maths Chapter 6 – Free PDF Download

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots are beneficial for students since it aids them in scoring high marks in the exam. The subject experts at BYJU’S outline the concepts in a distinct and well-defined fashion, keeping the IQ level of students in mind. These solution modules use various shortcut hints and practical examples to explain all the exercise questions in a simple and easily understandable language. Solving the NCERT Class 8 Solutions is a must to obtain an excellent score in the examination.

Here, with the NCERT Solutions provided, the students will learn various techniques to determine whether a given natural number is a perfect square or not. The answers to all problems within this chapter present inNCERT Textbooks are presented here in a detailed and step by step way to help the students understand more effectively. Students can download the NCERT Solutions for Class 8 Maths Chapter 6 from the link provided below.

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NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (1)
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NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (45)
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (46)

Access Answers of NCERT Class 8 Maths Chapter 6 – Squares and Square Roots

Exercise 6.1 Page: 96

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 a×a.

i. The unit digit of the square of a number having digit 1 as unit’s place is 1.

∴ 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 unit’s place is 4.

∴ 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 unit’s place is 1.

∴ 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 unit’s place is 9.

∴ 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 unit’s place is 6.

∴ 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 unit’s place is 9.

∴ 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 unit’s place is 4.

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

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (47)

viii. The unit digit of the square of a number having digit 0 as unit’s place is 01.

∴ 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 unit’s place is 6.

∴ 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 unit’s place is 5.

∴ 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:

We know that natural numbers ending in the digits 0, 2, 3, 7 and 8 are not perfect squares.

i. 1057 ⟹ Ends with 7

ii. 23453 ⟹ Ends with 3

iii. 7928 ⟹ Ends with 8

iv. 222222 ⟹ Ends with 2

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (48)

v. 64000 ⟹ Ends with 0

vi. 89722 ⟹ Ends with 2

vii. 222000 ⟹ Ends with 0

viii. 505050 ⟹ Ends with 0

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

i. 431

ii. 2826

iii. 7779

iv. 82004

Solution:

We know that the square of an odd number is odd and the square of an even number is even.

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. Observe the following pattern and find the missing numbers. 112 = 121

1012 = 10201

10012 = 1002001

1000012 = 1 …….2………1

100000012 = ……………………..

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.

∴ 1000012 = 10000200001

100000012 = 100000020000001

5. Observe the following pattern and supply the missing numbers. 112 = 121

1012 = 10201

101012 = 102030201

10101012 = ………………………

…………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, 10101012 =1020304030201

1010101012 =10203040505030201

6. Using the given pattern, find the missing numbers. 12 + 22 + 22 = 32

22 + 32 + 62 = 72

32 + 42 + 122 = 132

42 + 52 + _2 = 212

5 + _ 2 + 302 = 312

6 + 7 + _ 2 = _ 2

Solution:

Given, 12 + 22 + 22 = 32

i.e 12 + 22 + (1×2 )2 = ( 12 + 22 -1 × 2 )2

22 + 32 + 62 =72

∴ 22 + 32 + (2×3 )2 = (22 + 32 -2 × 3)2

32 + 42 + 122 = 132

∴ 32 + 42 + (3×4 )2 = (32 + 42 – 3 × 4)2

42 + 52 + (4×5 )2 = (42 + 52 – 4 × 5)2

∴ 42 + 52 + 202 = 212

52 + 62 + (5×6 )2 = (52+ 62 – 5 × 6)2

∴ 52 + 62 + 302 = 312

62 + 72 + (6×7 )2 = (62 + 72 – 6 × 7)2

∴ 62 + 72 + 422 = 432

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 = 144

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (49)

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

Solution:

We know, sum of first n odd natural numbers is n2 . Since,49 = 72

∴ 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 = 112

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

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (50)

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 n2 and (n+1)2, there are 2n non–perfect square numbers.

i. 122 and 132 there are 2×12 = 24 natural numbers.

ii. 252 and 262 there are 2×25 = 50 natural numbers.

iii. 992 and 1002 there are 2×99 =198 natural numbers.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (51)

Exercise 6.2 Page: 98

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 + 2×30×2 [Since, (a+b)2 = a2+b2 +2ab]

= 900 + 4 + 120

= 1024

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (52)

ii. (35)2

= (30+5 )2

= (30)2 + (5)2 + 2×30×5 [Since, (a+b)2 = a2+b2 +2ab]

= 900 + 25 + 300

= 1225

iii. (86)2

= (90 – 4)2

= (90)2 + (4)2 – 2×90×4 [Since, (a+b)2 = a2+b2 +2ab]

= 8100 + 16 – 720

= 8116 – 720

= 7396

iv. (93)2

(Video) Introduction - Squares and Square Roots, Chapter 6 - NCERT Class 8th Maths Solutions

= (90+3 )2

= (90)2 + (3)2 + 2×90×3 [Since, (a+b)2 = a2+b2 +2ab]

= 8100 + 9 + 540

= 8649

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (53)

v. (71)2

= (70+1 )2

= (70)2 + (1)2 +2×70×1 [Since, (a+b)2 = a2+b2 +2ab]

= 4900 + 1 + 140

= 5041

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (54)

vi. (46)2

= (50 -4 )2

= (50)2 + (4)2 – 2×50×4 [Since, (a+b)2 = a2+b2 +2ab]

= 2500 + 16 – 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, m2–1, m2+1 is a Pythagorean triplet.

i. 2m = 6

⇒ m = 6/2 = 3

m2–1= 32 – 1 = 9–1 = 8

m2+1= 32+1 = 9+1 = 10

∴ (6, 8, 10) is a Pythagorean triplet.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (55)

ii. 2m = 14

⇒ m = 14/2 = 7

m2–1= 72–1 = 49–1 = 48

m2+1 = 72+1 = 49+1 = 50

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

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (56)

iii. 2m = 16

⇒ m = 16/2 = 8

m2–1 = 82–1 = 64–1 = 63

m2+ 1 = 82+1 = 64+1 = 65

∴ (16, 63, 65) is a Pythagorean triplet.

iv. 2m = 18

⇒ m = 18/2 = 9

m2–1 = 92–1 = 81–1 = 80

m2+1 = 92+1 = 81+1 = 82

∴ (18, 80, 82) is a Pythagorean triplet.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (57)

Exercise 6.3 Page: 102

1. What could be the possible ‘one’s’ digits of the square root of each of the following numbers?

i. 9801

ii. 99856

iii. 998001

iv. 657666025

Solution:

i. We know that the unit’s digit of the square of a number having digit as unit’s

place 1 is 1 and also 9 is 1[92=81 whose unit place is 1].

∴ Unit’s digit of the square root of number 9801 is equal to 1 or 9.

ii. We know that the unit’s digit of the square of a number having digit as unit’s

place 6 is 6 and also 4 is 6 [62=36 and 42=16, both the squares have unit digit 6].

∴ Unit’s digit of the square root of number 99856 is equal to 6.

iii. We know that the unit’s digit of the square of a number having digit as unit’s

place 1 is 1 and also 9 is 1[92=81 whose unit place is 1].

∴ Unit’s digit of the square root of number 998001 is equal to 1 or 9.

iv. We know that the unit’s digit of the square of a number having digit as unit’s

place 5 is 5.

∴ Unit’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:

We know that natural numbers ending with the digits 0, 2, 3, 7 and 8 are not perfect square.

i. 153⟹ Ends with 3.

∴, 153 is not a perfect square

ii. 257⟹ Ends with 7

∴, 257 is not a perfect square

iii. 408⟹ Ends with 8

∴, 408 is not a perfect square

iv. 441⟹ Ends with 1

∴, 441 is a perfect square.

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

Solution:

100

100 – 1 = 99

99 – 3 = 96

96 – 5 = 91

91 – 7 = 84

84 – 9 = 75

75 – 11 = 64

64 – 13 = 51

51 – 15 = 36

36 – 17 = 19

19 – 19 = 0

Here, we have performed subtraction ten times.

∴ √100 = 10

169

169 – 1 = 168

168 – 3 = 165

165 – 5 = 160

160 – 7 = 153

153 – 9 = 144

144 – 11 = 133

133 – 13 = 120

120 – 15 = 105

105 – 17 = 88

88 – 19 = 69

69 – 21 = 48

48 – 23 = 25

25 – 25 = 0

Here, we have performed subtraction thirteen times.

∴ √169 = 13

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (58)

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.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (59)

729 = 3×3×3×3×3×3×1

⇒ 729 = (3×3)×(3×3)×(3×3)

⇒ 729 = (3×3×3)×(3×3×3)

⇒ 729 = (3×3×3)2

⇒ √729 = 3×3×3 = 27

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (60)

400 = 2×2×2×2×5×5×1

⇒ 400 = (2×2)×(2×2)×(5×5)

⇒ 400 = (2×2×5)×(2×2×5)

⇒ 400 = (2×2×5)2

⇒ √400 = 2×2×5 = 20

iii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (61)

1764 = 2×2×3×3×7×7

⇒ 1764 = (2×2)×(3×3)×(7×7)

⇒ 1764 = (2×3×7)×(2×3×7)

⇒ 1764 = (2×3×7)2

⇒ √1764 = 2 ×3×7 = 42

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (62)

iv.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (63)

4096 = 2×2×2×2×2×2×2×2×2×2×2×2

⇒ 4096 = (2×2)×(2×2)×(2×2)×(2×2)×(2×2)×(2×2)

⇒ 4096 = (2×2×2×2×2×2)×(2×2×2×2×2×2)

⇒ 4096 = (2×2×2×2×2×2)2

⇒ √4096 = 2×2×2 ×2×2×2 = 64

v.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (64)

7744 = 2×2×2×2×2×2×11×11×1

⇒ 7744 = (2×2)×(2×2)×(2×2)×(11×11)

⇒ 7744 = (2×2×2×11)×(2×2×2×11)

⇒ 7744 = (2×2×2×11)2

⇒ √7744 = 2×2×2×11 = 88

vi.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (65)

9604 = 62 × 2 × 7 × 7 × 7 × 7

(Video) Class 8 NCERT Maths Chapter 6 | Squares and Square Roots - Concepts, Examples & Practice Ques Solved

⇒ 9604 = ( 2 × 2 ) × ( 7 × 7 ) × ( 7 × 7 )

⇒ 9604 = ( 2 × 7 ×7 ) × ( 2 × 7 ×7 )

⇒ 9604 = ( 2×7×7 )2

⇒ √9604 = 2×7×7 = 98

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (66)vii.

5929 = 7×7×11×11

⇒ 5929 = (7×7)×(11×11)

⇒ 5929 = (7×11)×(7×11)

⇒ 5929 = (7×11)2

⇒ √5929 = 7×11 = 77

viii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (67)

9216 = 2×2×2×2×2×2×2×2×2×2×3×3×1

⇒ 9216 = (2×2)×(2×2) × ( 2 × 2 ) × ( 2 × 2 ) × ( 2 × 2 ) × ( 3 × 3 )

⇒ 9216 = ( 2 × 2 × 2 × 2 × 2 × 3) × ( 2 × 2 × 2 × 2 × 2 × 3)

⇒ 9216 = 96 × 96

⇒ 9216 = ( 96 )2

⇒ √9216 = 96

ix.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (68)

529 = 23×23

529 = (23)2

√529 = 23

x.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (69)

8100 = 2×2×3×3×3×3×5×5×1

⇒ 8100 = (2×2) ×(3×3)×(3×3)×(5×5)

⇒ 8100 = (2×3×3×5)×(2×3×3×5)

⇒ 8100 = 90×90

⇒ 8100 = (90)2

⇒ √8100 = 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.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (70)

252 = 2×2×3×3×7

= (2×2)×(3×3)×7

Here, 7 cannot be paired.

∴ We will multiply 252 by 7 to get perfect square.

New number = 252 × 7 = 1764

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (71)

1764 = 2×2×3×3×7×7

⇒ 1764 = (2×2)×(3×3)×(7×7)

⇒ 1764 = 22×32×72

⇒ 1764 = (2×3×7)2

⇒ √1764 = 2×3×7 = 42

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (72)

180 = 2×2×3×3×5

= (2×2)×(3×3)×5

Here, 5 cannot be paired.

∴ We will multiply 180 by 5 to get perfect square.

New number = 180 × 5 = 900

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (73)

900 = 2×2×3×3×5×5×1

⇒ 900 = (2×2)×(3×3)×(5×5)

⇒ 900 = 22×32×52

⇒ 900 = (2×3×5)2

⇒ √900 = 2×3×5 = 30

iii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (74)

1008 = 2×2×2×2×3×3×7

= (2×2)×(2×2)×(3×3)×7

Here, 7 cannot be paired.

∴ We will multiply 1008 by 7 to get perfect square.

New number = 1008×7 = 7056

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (75)

7056 = 2×2×2×2×3×3×7×7

⇒ 7056 = (2×2)×(2×2)×(3×3)×(7×7)

⇒ 7056 = 22×22×32×72

⇒ 7056 = (2×2×3×7)2

⇒ √7056 = 2×2×3×7 = 84

iv.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (76)

2028 = 2×2×3×13×13

= (2×2)×(13×13)×3

Here, 3 cannot be paired.

∴ We will multiply 2028 by 3 to get perfect square. New number = 2028×3 = 6084

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (77)

6084 = 2×2×3×3×13×13

⇒ 6084 = (2×2)×(3×3)×(13×13)

⇒ 6084 = 22×32×132

⇒ 6084 = (2×3×13)2

⇒ √6084 = 2×3×13 = 78

v.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (78)

1458 = 2×3×3×3×3×3×3

= (3×3)×(3×3)×(3×3)×2

Here, 2 cannot be paired.

∴ We will multiply 1458 by 2 to get perfect square. New number = 1458 × 2 = 2916

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (79)

2916 = 2×2×3×3×3×3×3×3

⇒ 2916 = (3×3)×(3×3)×(3×3)×(2×2)

⇒ 2916 = 32×32×32×22

⇒ 2916 = (3×3×3×2)2

⇒ √2916 = 3×3×3×2 = 54

vi.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (80)

768 = 2×2×2×2×2×2×2×2×3

= (2×2)×(2×2)×(2×2)×(2×2)×3

Here, 3 cannot be paired.

∴ We will multiply 768 by 3 to get perfect square.

New number = 768×3 = 2304

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (81)

2304 = 2×2×2×2×2×2×2×2×3×3

⇒ 2304 = (2×2)×(2×2)×(2×2)×(2×2)×(3×3)

⇒ 2304 = 22×22×22×22×32

⇒ 2304 = (2×2×2×2×3)2

⇒ √2304 = 2×2×2×2×3 = 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

Solution:

i.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (82)

252 = 2×2×3×3×7

= (2×2)×(3×3)×7

Here, 7 cannot be paired.

∴ We will divide 252 by 7 to get perfect square. New number = 252 ÷ 7 = 36

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (83)

36 = 2×2×3×3

⇒ 36 = (2×2)×(3×3)

⇒ 36 = 22×32

⇒ 36 = (2×3)2

⇒ √36 = 2×3 = 6

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (84)

2925 = 3×3×5×5×13

= (3×3)×(5×5)×13

Here, 13 cannot be paired.

∴ We will divide 2925 by 13 to get perfect square. New number = 2925 ÷ 13 = 225

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (85)

225 = 3×3×5×5

⇒ 225 = (3×3)×(5×5)

⇒ 225 = 32×52

⇒ 225 = (3×5)2

⇒ √36 = 3×5 = 15

iii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (86)

396 = 2×2×3×3×11

= (2×2)×(3×3)×11

Here, 11 cannot be paired.

∴ We will divide 396 by 11 to get perfect square. New number = 396 ÷ 11 = 36

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (87)

36 = 2×2×3×3

⇒ 36 = (2×2)×(3×3)

⇒ 36 = 22×32

⇒ 36 = (2×3)2

⇒ √36 = 2×3 = 6

iv.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (88)

2645 = 5×23×23

⇒ 2645 = (23×23)×5

(Video) Class 8 Maths Chapter 6 | Squares and Square Roots Full Chapter Explanation & Exercise (Part 1)

Here, 5 cannot be paired.

∴ We will divide 2645 by 5 to get perfect square.

New number = 2645 ÷ 5 = 529

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (89)

529 = 23×23

⇒ 529 = (23)2

⇒ √529 = 23

v.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (90)

2800 = 2×2×2×2×5×5×7

= (2×2)×(2×2)×(5×5)×7

Here, 7 cannot be paired.

∴ We will divide 2800 by 7 to get perfect square. New number = 2800 ÷ 7 = 400

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (91)

400 = 2×2×2×2×5×5

⇒ 400 = (2×2)×(2×2)×(5×5)

⇒ 400 = (2×2×5)2

⇒ √400 = 20

vi.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (92)

1620 = 2×2×3×3×3×3×5

= (2×2)×(3×3)×(3×3)×5

Here, 5 cannot be paired.

∴ We will divide 1620 by 5 to get perfect square. New number = 1620 ÷ 5 = 324

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (93)

324 = 2×2×3×3×3×3

⇒ 324 = (2×2)×(3×3)×(3×3)

⇒ 324 = (2×3×3)2

⇒ √324 = 18

7. The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister’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.

∴ Each student donate Rs.x .

Total many contributed by all the students= x×x=x2 Given, x2 = Rs.2401

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (94)

x2 = 7×7×7×7

⇒ x2 = (7×7)×(7×7)

⇒ x2 = 49×49

⇒ x = √(49×49)

⇒ x = 49

∴ The number of students = 49

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (95)

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.

∴ the number of plants in each rows = x.

Total many contributed by all the students = x × x =x2

Given,

x2 = Rs.2025

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (96)

x2 = 3×3×3×3×5×5

⇒ x2 = (3×3)×(3×3)×(5×5)

⇒ x2 = (3×3×5)×(3×3×5)

⇒ x2 = 45×45

⇒ x = √45×45

⇒ x = 45

∴ 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:

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (97)

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

180 = 2×2×9×5

= (2×2)×3×3×5

= (2×2)×(3×3)×5

Here, 5 cannot be paired.

∴ we will multiply 180 by 5 to get perfect square.

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

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

Solution:

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (98)

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

120 = 2×2×3×5×2

= (2×2)×3×5×2

Here, 3, 5 and 2 cannot be paired.

∴ We will multiply 120 by (3×5×2) 30 to get perfect square.

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

Exercise 6.4 Page: 107

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.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (99)

∴ √2304 = 48

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (100)

∴ √4489 = 67

iii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (101)

∴ √3481 = 59

iv.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (102)

∴ √529 = 23

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (103)v.

∴ √3249 = 57

vi.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (104)

∴ √1369 = 37

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (105)vii.

∴ √5776 = 76

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (106)viii.

∴ √7921 = 89

ix.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (107)

∴ √576 = 24

x.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (108)

∴ √1024 = 32

xi.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (109)

∴ √3136 = 56

xii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (110)

∴ √900 = 30

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

calculation).64

i. 144

ii. 4489

iii. 27225

iv. 390625

Solution:

i.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (111)

∴ √144 = 12

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

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (112)

∴ √4489 = 67

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

iii.
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (113)

√27225 = 165

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

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (114)iv.

∴ √390625 = 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.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (115)

∴ √2.56 = 1.6

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (116)

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (117)

∴ √7.29 = 2.7

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (118)

iii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (119)

∴ √51.84 = 7.2

(Video) Class 8 NCERT Maths Chapter 6 | Squares and Square Roots - Exercise 6.1 Solutions

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (120)

iv.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (121)

∴ √42.25 = 6.5

(v)
NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (122)v.

∴ √31.36 = 5.6

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (123)

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.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (124)∴ √400 = 20

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

New number = 402 – 2 = 400

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (125)

∴ √400 = 20

ii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (126)

∴ We must subtracted 53 from 1989 to get a perfect square. New number = 1989 – 53 = 1936

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (127)

∴ √1936 = 44

iii.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (128)

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

New number = 3250 – 1 = 3249

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (129)

∴ √3249 = 57

iv.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (130)

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

New number = 825 – 41 = 784

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (131)

∴ √784 = 28

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (132)

∴ We must subtracted 31 from 4000 to get a perfect square. New number = 4000 – 31 = 3969

∴ √3969 = 63

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)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (133)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (134)

Here, (22)2 < 525 > (23)2

We can say 525 is ( 129 – 125 ) 4 less than (23)2.

∴ If we add 4 to 525, it will be perfect square. New number = 525 + 4 = 529

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (135)

∴ √529 = 23

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (136)(ii)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (137)

Here, (41)2 < 1750 > (42)2

We can say 1750 is ( 164 – 150 ) 14 less than (42)2.

∴ If we add 14 to 1750, it will be perfect square.

New number = 1750 + 14 = 1764

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (138)

∴√1764 = 42

(iii)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (139)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (140)

Here, (15)2 < 252 > (16)2

We can say 252 is ( 156 – 152 ) 4 less than (16)2.

∴ If we add 4 to 252, it will be perfect square.

New number = 252 + 4 = 256

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (141)

∴ √256 = 16

(iv)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (142)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (143)

Here, (42)2 < 1825 > (43)2

We can say 1825 is ( 249 – 225 ) 24 less than (43)2.

∴ If we add 24 to 1825, it will be perfect square.

New number = 1825 + 24 = 1849

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (144)

∴ √1849 = 43

(v)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (145)

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (146)

Here, (80)2 < 6412 > (81)2

We can say 6412 is ( 161 – 12 ) 149 less than (81)2.

∴ If we add 149 to 6412, it will be perfect square.

New number = 6412 + 149 = 656

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (147)

∴ √6561 = 81

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

Solution:

Let the length of each side of the field = a Then, area of the field = 441 m2

⇒ a2 = 441 m2

⇒a = √441 m

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (148)

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

7. In a right triangle ABC, ∠B = 90°.

a. If AB = 6 cm, BC = 8 cm, find AC

b. If AC = 13 cm, BC = 5 cm, find AB

Solution:

a.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (149)

Given, AB = 6 cm, BC = 8 cm

Let AC be x cm.

∴ AC2 = AB2 + BC2

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (150)

Hence, AC = 10 cm.

b.

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (151)

Given, AC = 13 cm, BC = 5 cm

Let AB be x cm.

∴ AC2 = AB2 + BC2

⇒ AC2 – BC2 = AB2

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (152)

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.

∴ Total number of row and column= x× x = x2 As per question, x2 = 1000

⇒ x = √1000

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (153)

Here, (31)2 < 1000 > (32)2

We can say 1000 is ( 124 – 100 ) 24 less than (32)2.

∴ 24 more plants are 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.

∴ Total number of row and column= x × x = x2 As per question, x2 = 500

x = √500

NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots (154)

Hence, 16 children would be left out in the arrangement

Also Access
NCERT Exemplar for class 8 Maths Chapter 6
CBSE Notes for class 8 Maths Chapter 6

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

The NCERT Solutions for Class 8 MathsChapter 8 deals with the concept of squares and square roots, along with other main topics and concepts.

The major concepts covered in this chapter include:

6.1 Introduction

6.2 Properties of Square Numbers

6.3 Some interesting patterns

6.4 Finding the square of a number

6.4.1 Patterns in squares

6.4.2 Pythagorean triplets

6.5 Square Roots

6.5.1 Finding square roots

6.5.2 Finding square root through repeated subtraction

6.5.3 Finding square root through prime factorisation

6.5.4 Finding square root by division method

6.6 Square Roots of Decimals

6.7 Estimating Square Root

Below are the exercise-wise NCERT Solutions for Class 8 Maths Chapter 6:

Exercise 6.1 Solutions 9 Questions

Exercise 6.2 Solutions 2 Questions

Exercise 6.3 Solutions 10 Questions

Exercise 6.4 Solutions 9 Questions

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

Chapter 6 of NCERT Solutions for Class 8 Maths discusses the following:

  1. If a natural number m can be expressed as n2, where n is a natural number, then m is a square number.
  2. All square numbers end with 0, 1, 4, 5, 6 or 9 at units place.
  3. Square numbers can only have an even number of zeros at the end.
  4. Square root is the inverse operation of square.
  5. There are two integral square roots of a perfect square number

Learning the chapter Squares and Square Roots enables the students to understand:

  • Square and Square roots
  • Square roots using factor method and division method for numbers containing(a) no more than a total of 4 digits and (b) no more than 2 decimal places
  • Estimating square roots
  • Learning the process of moving nearer to the required number

Disclaimer:

Dropped Topics – 6.7 Estimating square roots.

Frequently Asked Questions on NCERT Solutions for Class 8 Maths Chapter 6

Is BYJU’S website providing answers for NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots?

Yes, you can avail the PDFs of NCERT Solutions for Class 8 Maths Chapter 6. These solutions are formulated by expert faculty at BYJU’S in a unique way. Also, they provide solutions for Class 1 to 12 NCERT Textbooks in free PDFs. Who wish to score high in exams are advised to solve the NCERT Textbook.

Mention the important concepts that you learn in NCERT Solutions for Class 8 Maths Chapter 6 Squares and Square Roots?

The concepts presented in NCERT Solutions for Class 10 Maths Chapter 6 are listed here:
6.1 Introduction
6.2 Properties of Square Numbers
6.3 Some interesting patterns
6.4 Finding the square of a number
6.4.1 Patterns in squares
6.4.2 Pythagorean triplets
6.5 Square Roots
6.5.1 Finding square roots
6.5.2 Finding square root through repeated subtraction
6.5.3 Finding square root through prime factorisation
6.5.4 Finding square root by division method
6.6 Square Roots of Decimals
6.7 Estimating Square Root

(Video) Chapter 6 Square and Square Roots || Full Exercise 6.1 & Basic || Class 8 Maths RBSE CBSE NCERT

How many exercises are there in NCERT Solutions for Class 8 Maths Chapter 6?

The sixth Chapter of NCERT Solutions for Class 8 Maths has 4 exercises. They are
Exercise 6.1 with 9 Questions
Exercise 6.2 with 2 Questions
Exercise 6.3 with 10 Questions
Exercise 6.4 with 9 Questions

Videos

1. Chapter:6 (Intro)+Ex 6 1 (Q.1,2,3) Squares and Square Roots | Ncert | Maths | Class 8 | Cbse.
(Ranveer Maths 8)
2. Class 8 Squares and Square Roots Exercise 6.3 Part1-Q#1-4 (In English)- NCERT CBSE
(Divya's Maths Solutions)
3. try these - page 92 -- chapter 6 - Squares and Square Roots - class 8 - maths - ncert
(KNOWLEDGE IS POWER)
4. Class 8 NCERT Maths Chapter 6 | Squares and Square Roots - Exercise 6.3 Solutions
(Magnet Brains)
5. Q 1 - Ex 6.1 - Square and Square Roots - NCERT Maths Class 8th - Chapter 6
(Mathematics Class 8)
6. Class 8 NCERT Maths Ch 6 | Squares and Square Roots - Concepts, Examples & Exercise 6.4 Solutions
(Magnet Brains)

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