**NCERT Solutions for Class 8 Maths Chapter 5** Squares and Square Roots are beneficial for the students and important for the next class as it helps them to score higher marks in the examination.

The subject experts at Study Circle outline the concepts in a specific and well-defined manner keeping in mind the IQ level of the students.

These solutions are a fair attempt to make all the questions simple and easy to understand. Solving **NCERT Solution for Class 8 Maths Chapter 5** is essential to score excellent marks in the exam. This chapter is based on the new syllabus.

**NCERT Solutions for Class 8 Maths Chapter 5**

**Exercise 5.1**

**Question 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.

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.

**Question 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

v. 64000 ⟹ Ends with 0

vi. 89722 ⟹ Ends with 2

vii. 222000 ⟹ Ends with 0

viii. 505050 ⟹ Ends with 0

**Question 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.

**Question 4. Observe the following pattern and find the missing numbers. 11 ^{2} = 121**

**101 ^{2} = 10201**

**1001 ^{2} = 1002001**

**100001 ^{2} = 1 …….2………1**

**10000001 ^{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.

∴ 100001^{2} = 10000200001

10000001^{2} = 100000020000001

**Question 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

**Question 6. 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} + (1×2 )^{2} = ( 1^{2} + 2^{2} -1 × 2 )^{2}

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

∴ 2^{2} + 3^{2} + (2×3 )^{2} = (2^{2} + 3^{2} -2 × 3)^{2}

3^{2 }+ 4^{2} + 12^{2} = 13^{2}

∴ 3^{2} + 4^{2} + (3×4 )^{2} = (3^{2} + 4^{2} – 3 × 4)^{2}

4^{2} + 5^{2} + (4×5 )^{2} = (4^{2} + 5^{2} – 4 × 5)^{2}

∴ 4^{2} + 5^{2} + 20^{2} = 21^{2}

5^{2} + 6^{2} + (5×6 )^{2} = (5^{2}+ 6^{2} – 5 × 6)^{2}

∴ 5^{2} + 6^{2} + 30^{2} = 31^{2}

6^{2} + 7^{2} + (6×7 )^{2} = (6^{2} + 7^{2} – 6 × 7)^{2}

∴ 6^{2} + 7^{2} + 42^{2} = 43^{2}

**Question 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

**Question 8. (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}

∴ 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}

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

**Question 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 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 5**

**Exercise 5.2**

**Question 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} = a^{2}+b^{2} +2ab]

= 900 + 4 + 120

= 1024

**(ii) ** (35)^{2}

= (30+5 )^{2}

= (30)^{2} + (5)^{2} + 2×30×5 [Since, (a+b)^{2} = a^{2}+b^{2} +2ab]

= 900 + 25 + 300

= 1225

**(iii) ** (86)^{2}

= (90 – 4)^{2}

= (90)^{2} + (4)^{2} – 2×90×4 [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} + 2×90×3 [Since, (a+b)^{2} = a^{2}+b^{2} +2ab]

= 8100 + 9 + 540

= 8649

**( v)** (71)^{2}

= (70+1 )^{2}

= (70)^{2} + (1)^{2} +2×70×1 [Since, (a+b)^{2} = a^{2}+b^{2} +2ab]

= 4900 + 1 + 140

= 5041

**(vi)** (46)^{2}

= (50 -4 )^{2}

= (50)^{2} + (4)^{2} – 2×50×4 [Since, (a+b)^{2} = a^{2}+b^{2} +2ab]

= 2500 + 16 – 400

= 2116

**Question 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.

**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.

**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 5**

**Exercise 5.3 **

**Question 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[9^{2}=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 [6^{2}=36 and 4^{2}=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[9^{2}=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.

**Question 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.

**Question 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

**Question 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 = 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)**

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) 1764**

**(iv) 4096**

**(v) 7744**

**(vi) 9604**

9604 = 62 × 2 × 7 × 7 × 7 × 7

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

⇒ 9604 = ( 2 × 7 ×7 ) × ( 2 × 7 ×7 )

⇒ 9604 = ( 2×7×7 )^{2}

⇒ √9604 = 2×7×7 = 98

**(vii) 5929**

**(viii) 9216**

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) 529**

**(x) 8100**

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

**Question 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:–**

**(ii) 180 **

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

900 = 2×2×3×3×5×5×1

⇒ 900 = (2×2)×(3×3)×(5×5)

⇒ 900 = 2^{2}×3^{2}×5^{2}

⇒ 900 = (2×3×5)^{2}

⇒ √900 = 2×3×5 = 30

**(iii) 1008 **

**(iv) 2028 **

**(v) 1458 **

**(vi) 768 **

**Question 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) **

**(ii) 2925 **

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

225 = 3×3×5×5

⇒ 225 = (3×3)×(5×5)

⇒ 225 = 3^{2}×5^{2}

⇒ 225 = (3×5)^{2}

⇒ √36 = 3×5 = 15

**(iii) 396 **

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

36 = 2×2×3×3

⇒ 36 = (2×2)×(3×3)

⇒ 36 = 2^{2}×3^{2}

⇒ 36 = (2×3)^{2}

⇒ √36 = 2×3 = 6

**(iv) 2645 **

**(v) 2800 **

**(vi) 1620 **

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

324 = 2×2×3×3×3×3

⇒ 324 = (2×2)×(3×3)×(3×3)

⇒ 324 = (2×3×3)^{2}

⇒ √324 = 18

**Question 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 amount contributed by all the students= x×x=x^{2} Given, x^{2} = Rs.2401

x^{2} = 7×7×7×7

⇒ x^{2} = (7×7)×(7×7)

⇒ x^{2 }= 49×49

⇒ x = √(49×49)

⇒ x = 49

∴ The number of students = 49

**Question 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 plants to be planted in the garden = x × x =x^{2}

Given,

x_{2} = Rs.2025

x^{2} = 3×3×3×3×5×5

⇒ x^{2} = (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.

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

**Solution:**

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:–**

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

**NCERT Solutions for Class 8 Maths Chapter 5 **

**Exercise 5.4**

**Question 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) **

**(ii) 4489 **

**(iii) 3481**

**(iv) 529 **

**(v) 3249 **

**(vi) 1369 **

**(vii) 5776 **

**(viii) 7921**

**(ix) 576**

**(x) 1024 **

**(ix) 3136**

**(xii) 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**

**Solution:–**

**(i) **Here, 64 contains two digits which is even.

Therefore, number of digits in square root =

**(ii)** Here, 144 contains three digits which is odd.

Therefore, number of digits in square root =

**(iii)** Here, 4489 contains four digits which is even.

Therefore, number of digits in square root =

**(iv)** Here, 27225 contains fivr digits which is odd.

Therefore, number of digits in square root =

**(v)** Here, 390625 contains six digits which is even.

Therefore, number of digits in square root =

**Question 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) **

**(ii)**

** **

**(iii)**

**(iv) **

**(v) **

**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**

**ii. 1989**

**iii. 3250**

**iv. 825**

**v. 4000**

**Solution:–(i)**

**(ii) **

**(iii) **

**(iv) **

**(v) **

**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**

**(ii) 1750**

**(iii) 252**

**(iv)1825**

**(v)6412**

**Solution:– (i) **

**(ii) 1750**

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

∴√1764 = 42

**(iii) 252 **

**(iv) 1825 **

**(v) 6412 **

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

∴ √6561 = 81

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

**Solution:–**

**Question 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:–**

**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.**

**Solution:–**

**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.**

**Solution:–**