**Rd Sharma Solutions Class 8 Chapter 3 Squares and Square Roots**

Welcome to NCTB Solution. Here with this post we are going to help 8th class students for the Solutions of RD Sharma Class 8 Mathematics, Chapter 3, Squares and Square Roots. Here students can easily find Exercise wise solution for chapter 3, Squares and Square Roots. Students will find proper solutions for Exercise 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9 Our teacher’s solved every problem with easily understandable methods so that every students can understand easily.

**Squares and Square Roots Exercise 3.1 Solution : **

**Question no – (1) **

**Solution : **

**(i) 484**

= Not Perfect Square.

**(ii) 625**

(25 × 25)

**∴** It is perfect square.

**(iii) 576**

= (24 × 24)

**∴** It is perfect square

**(iv) 941**

= Not perfect square

**(v) 961**

= (31 × 31)

**∴** it is perfect square

**(vi) 2500**

= (50 × 50)

**∴** It is perfect square.

**Question no – (2) **

**Solution : **

**(i) 1156**

= 2 × 1 × 17 × 17

= 2 × 17

= 34

**(ii) 2025**

= 3 × 3 × 3 × 3 × 5 × 5

= 3 × 3 × 5

= 45

**(iii) 14611**

= 11 × 11 × 11 × 11

= 11 × 11

= 212

**(iv) 4761**

= 3 × 3 × 23 × 23

= 3 × 23

= 69

**Question no – (3) **

**Solution : **

**(i) 23805**

= (3 × 3) (23 × 23) × 5

**∴** 5 should be multiplied.

**(ii) 12150**

= 2 × (3 × 3) × (3 × 3) × (5 × 5)

**∴** 2 should be multiplied.

**(iii) 7688**

= (31 × 31) × (2 × 2) × 2

**∴** 2 should be multiplied.

**Question no – (4) **

**Solution : **

**(i) 14283**

= (23 × 23) × (3 × 3) × 3

**∴** 3 should be divided

**(ii) 1800**

= (5 × 5) × (3 ×) × (2 × 2) × 2

**∴** 2 should be divided

**(iii) 2904**

= (11 × 11) × (2 × 2) × 2 × 3

**∴** 6 should be divided.

**Question no – (5) **

**Solution : **

From the above numbers the perfect square number are –

16 = 4 × 4

36 = 6 × 6

64 = 8 × 8

81 = 9 × 9

121 = 11 × 11

**Question no – (6) **

**Solution : **

The perfect square numbers are –

225 = 5 × 5 × 3 × 3

441 = 3 × 3 × 7 × 7

2916 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

11025 = 5 × 5 v 3 × 3 × 7 × 7

**Question no – (7) **

**Solution : **

**(i) 8820**

= (3 × 3) × (7 × 7) × (2 × 2) × 5

**∴** 5 should be multiplied to get a perfect square.

**∴** Required perfect square number = 44100 = 210 × 210

**(ii) 3675**

= (5 × 5) × 3 × (7 × 7)

**∴** 3 should be multiplied

**∴** Required perfect square number = 11025 = 105 × 105

**(iii) 605**

= (11 × 11) × (5 × 5) × 5

**∴** 5 should be multiplied

**∴** Required perfect square number = 3025 = 55 × 55

**(iv) 2880**

= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5

**∴** 5 should be multiplied

**∴** Required perfect square number = 14400 = 120 × 120

**(v) 4056**

= (2 × 2) × (13 × 13) × 2 × 3

**∴** 6 should be multiplied

**∴** Required perfect square number = 24336 = 156 × 156

**(vi) 3468**

= (2 × 2) × (17 × 17) × 3

**∴** 3 should be multiplied

**∴** Required perfect square number = 10404 = 102 × 102

**(vii) 7776**

= (3 × 3) × (3 × 3) × 3 × (2 × 2) × (2 × 2) × (2 × 2) × 2

**∴** 6 should be perfect multiplied

**∴** Required perfect square number

= 46656

= 216 × 216

**Question no – (8) **

**Solution : **

**(i) 16562**

= (7 × 7) × (13 × 13) × 2

**∴** 2 should be divided

**∴** Required perfect square number

= 8281

= 91 × 91

**(ii) 3698**

= (43 × 43) × 2

**∴** 2 should be divided

**∴** Required perfect square number= 1849 = 43 × 43

**(iii) 5103**

= (3 × 3) × (3 × 3) × (3 × 3) × 7

**∴** 7 should be divided

**∴** Required perfect square number = 729 = 27 × 27

**(iv) 3174**

= (23 × 23) × 2 × 3

**∴** 6 should be divided

**∴** Required perfect square number = 529 = 23 × 23

**(v) 1575**

= (3 × 3) × (5 × 5) × 7

**∴** 7 should be divided

**∴** Required perfect square number = 225 = 15 × 15

**Question no – (9) **

**Solution : **

The greatest number of two digits which is a perfect square is – 81 = 9 × 9

**Question no – (10) **

**Solution : **

The least number of three digits which is perfect square is –

100 = 10 × 10

**Question no – (11) **

**Solution : **

Given number, 4851

= 3 × 3 × 7 × 7 × 11

**∴** 11 should be multiplied.

**∴** Required perfect square numbers,

= 3 × 3 × 7 × 7 × 11 × 11

= 53361

**Question no – (13) **

**Solution : **

Given number,

= 1152

= 3 × 3 × 2 × 2 × 2 × 2 × 2 × 2 × 2

**∴** 2 should be divided.

**∴** Resulting square number

= 576

= 24 × 24

**Squares and Square Roots Exercise 3.2 Solution : **

**Question no – (1) **

**Solution : **

**(i) 1547**

= 7 × 13 × 17

**∴** There are no pair

**∴** it is not perfect square.

**(ii) 45743**

= 149 × 307

**∴** There are no pair

**∴** it is not perfect square.

**(iii) 8943**

= 271 × 2 × 2 × 2

**∴** There are many number without pair.

**∴** it is not perfect square.

**(iv) 333333**

= 3 × 3 × 11 × 7 × 37 × 13

**∴** There are many number without pair.

**∴** it is not perfect square.

**Question no – (3) **

**Solution : **

**(i) 731**

= odd number

**(ii) 4058**

= even number

**(iii) 5559**

= odd number

**(iv) 42008**

= even number.

**Question no – (4) **

**Solution : **

**(i) 52**

= unit digit

= 2²

= 4

**(ii) 977**

= unit digit

= 7² = 4 [9]

**(iii) 4583**

= unit digit

= 3²

= 9

**(iv) 78367**

= unit digit

= 7² 4 [9]

**(v) 52698**

= unit digit

= 8²

= 6 [4]

**(vi) 99880**

= unit digit

= 0

**(vii) 12796**

= unit digit

= unit digit

= 6²

= 3 [6]

**(viii) 55555**

= unit digit

= 5²

= 2 [5]

**(ix) 53924**

= unit digit

= unit digit

= 4²

= 1 [6]

**Question no – (7) **

**Solution : **

**(i) 100² – 99²**

= 100 + 99

= 199

**(ii) 111² – 109²**

= 111 + 109

= 219

**(iii) 99² – 96²**

= 99 + 96

= 195

**Question no – (7) **

**Solution : **

**(i) (8, 15, 17)**

= (15)² + (8)² = (17)²

= 289 = 289

**∴** It is Pythagorean Triplets

**(ii) (18, 80, 82)**

= (18)² + (80)² = (82)²

= 6724 = 8724

**∴** It is Pythagorean Triplets

**(iii) (14, 48, 5)**

= (14)² + (48)² = (51)²

= 2000 ≠ 2601

**∴** It is not Pythagorean Triplets

**(iv) (10, 24, 26)**

= (10)² + (24)² = (26)²

= 676 = 676

**∴ **It is Pythagorean Triplets

**(v) (16, 63, 65)**

= (16)² + (63)² = (65)²

= 4225 = 4225

**∴** It is Pythagorean Triplets

**(vi) (12, 35, 38)**

= (2)² + (35)² = (38)²

= 1369 ≠ 1444

**∴ **It is not Pythagorean Triplets.

**Question no – (8) **

**Solution : **

Given numbers,

= (1 × 2) + (2 × 3) + (3 × 4) + (4 × 5) + (5 × 6)

= 5 × 6 × 7/ 3

= 210/3

= 70

Therefore, the value will be 70.

**Question no – (9) **

**Solution : **

**(i) 1 + 2 + 3 + 4 + 5 + … + 50**

= 1/2 {50 × (50 + 1)}

= 1/2 × 50 × 51

= 1275

**(ii) 31 + 32 + … + 50**

= An = 50

a = 31

Common difference = d = 32 – 31 = 1

**∴** An = a + (n – 1) × d

or, 50 = 31 + (n – 1) × 1

or, (n – 1) = 50 – 31 = 19

or, n = 19 + 1 = 20

**∴** Sn = n/2 (2a + (n-1) d)

= 20/2 (2 × 31 + (20 – 1) 1)

= 10 [62 + 10]

= 10 × 81 = 810

**Question no – (11) **

**Solution : **

The squares of even numbers are, 256, 324, 1296, 5476

**Question no – (12) **

**Solution : **

**(i) 1026**

= unit digit

= 6 cannot be square.

**(ii) 1028**

= unit digit

= 8 = It may be a square.

**(iii) 1024**

= unit digit

= 2 = It may be a square.

**(iv) 1022**

= unit digit

= 3

∴ It cannot be perfect square.

**(v) 1023**

= unit digit

= 3

∴ It cannot be perfect square.

**(vi) 1027**

= unit digit

= 7

∴ It cannot be a perfect.

**Question no – (13) **

**Solution : **

The five numbers are,

(i) 12461

(ii) 5680

(iii) 1734

(iv) 915389

(v) 25247

**Question no – (14) **

**Solution : **

When unit digit of a number end with ‘0’ ‘7’ ‘3’ ‘2’ then it will be not square. Example – 520, 47, 1753, 4122, 1980

**Question no – (15) **

**Solution : **

**(i)** The number of digits in a perfect square is even – **(False).**

**(ii)** The square of a prime number is prime – **(False).**

**(iii)** The sum of two square numbers is a square number – **(False).**

**(vi)** No square number is negative – **(True).**

**(vii)** There is no square number between 50 and 60 – **(True).**

**(viii)** There are fourteen square number up to 200 – **(True).**

**Squares and Square Roots Exercise 3.3 Solution : **

**Question no – (1) **

**Solution : **

**(i) 25 = a = 2, b = 5**

Col – I |
Col – II |
Col – III |

a ² | 2 × a × b | b² |

2 ² | 2 × 2 × 5 | 5² |

4 | 20 | 25 |

+2 | + 2 | 5 |

6 | 22 | |

2 |

**∴** (25)² = 625

**(ii) 37 = a = 3 b = 7**

Col – I |
Col – II |
Col – III |

a ² | 2 × a × b | b² |

3² | 2 × 3 × 7 | 7² |

9 | 42 | 49 |

+ 4 | + 4 | 9 |

13 | 46 | |

6 |

**∴** (37)² = 1369

**(iii) 54 = a = 5 = b = 4**

Col – I |
Col – II |
Col – III |

a² | 2 × a × b | b² |

5² | 2 × 5 × 4 | 4² |

25 | 40 | 16 |

+ 4 | + 1 | 6 |

29 | 41 | |

1 |

**∴** (54)² = 2916

**(iv) 71 = a = 7, b = 1**

Col – I |
Col – II |
Col – III |

a² | 2 × a × b | b² |

7² | 2 × 7 × 1 | 1² |

49 | 14 | 1 |

+ 1 | + 0 | 1 |

50 | 14 | |

4 |

**∴** (71)² = 5041

**(v) 96 = a = 9 b = 6**

Col – I |
Col – II |
Col – III |

a² | 2 × a × b | b² |

9² | 2 × 9 × 6 | 6² |

81 | 108 | 36 |

+ 11 | + 3 | 6 |

92 | 111 | |

1 |

**∴** (96)² = 9216

**Question no – (2) **

**Solution : **

**(i) 98**

∴ (98)² = 9606

**(ii) 273**

**∴** (273)² = 7452

**(iii) 348**

**∴** (348)² = 121104

**(iv) 295**

**∴** (295)² = 87025

**(v) 171**

**∴ **(171)² = 29241

**Question no – (4)**

**Solution : **

**(i) 425
**

= (425)²

= 425 ×425

= 180625

**(ii) 575
**

= (575)²

= 575 × 575

= 330625

**(iii) 405
**

= (405)²

= 405 × 405

= 164025

**(iv) 205
**

= (205)²

= 205 × 205

= 42025

**(v) 95**

= (95)²

= 95 × 95

= 9025

**(vi) 745
**

= (745)²

= 745 × 745

= 555025

**(vii) 512**

= (512)²

= 512 × 512

= 262144

**(viii) 995**

= (995)²

= 995 × 995

= 990025

**Question no – (5) **

**Solution : **

**(i) 405**

405 = (400 + 5)²

= (400) ² + 2.400. 5 + 5²

= 160000 + 4000 + 25

= 164025

**(ii) 510**

510 = (500 + 10)²

= (500) ² + 2. 500 .10 + (10) ²

= 250000 + 10000 + 100

= 260100

**(iii) 1001**

= 1001 = (100 + 1)²

= (1000) ² + 2. 1000. 1 + 1²

= 1000000 + 2000 + 1

= 1002001

**(iv) 209**

209 = (200 + 9)²

= (200) ² + 2. 200.9 + 9²

= 40000 + 3600 + 81

= 43681

**(v) 605**

605 = (600 + 5)² (600)² 2. 600. 5 + 5²

= 360000 + 6000 + 25

= 366025

**Question no – (7) **

**Solution : **

**(i) 52**

**∴** (52)² = 2500 + 100 + 100 + 4

= 2704

**(ii) 95**

**∴** (95)² = 8100 + 450 + 450 = 25

= 9025

**(iii) 505**

**∴** (505)² = 250000 + 2500 + 2500 + 25

= 255025

**(iv) 702**

**∴** (702)2 = 490000 + 1400 + 1400 + 4

= 492804

**(v) 99**

**∴** (99)² = 8100 + 810 + 810 + 36

= 9801

**Squares and Square Roots Exercise 3.4 Solution : **

**Question no – (1) **

**Solution : **

**(i) 9801**

= unit digit 1 → 1 or 9

**∴** odd square root

**(ii) 99856**

= unit digit 1 →1 or 6

**∴** odd square root

**(iii) 998001**

= unit digit 1 → 1 or 9

**∴** odd square root

**(iv) 6576625**

= unit digit 5 → 5

∴ odd square root.

**Question no – (2) **

**Solution : **

**(i) 441**

= √441 = √ 3 × 3 × 7 × 7

= 3 × 7

= 21

**(ii) 196**

= √196 = √2 × 2 × 7 × 7

= 2 × 7

= 14

**(iii) 529**

√529 = √23 × 23

= 23

**(iv) 1764**

= √1764 = √2 × 2 × 3 × 3 × 7 × 7

= 2 × 3 × 7

= 42

**(v) 1156**

= √1156 = √2 × 2 × 17 × 17

= 2 × 17

= 34

**(vi) 4096**

= √4096 = √2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 2 × 2 × 2 × 2 × 2 × 2

= 64

**(vii) 7056**

= √7056 = √2 × 2 × 2 × 2 × 3 × 3 × 7 × 7

= 2 × 2 × 3 × 7

= 84

**(viii) 8281**

= √8281 = √7 × 7 × 13 × 13

= 7 × 13

= 91

**(ix) 11664**

= √11614 = √2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

= 2 × 2 × 3 × 3 × 3

= 108

**(x) 47089**

= √47089 = √7 × 7 × 31 × 31

= 7 × 13

= 217

**(xi) 24336**

= √24336 = √2 × 2 × 2 × 2 × 3 × 3 × 13 × 13

= 2 × 2 × 3 × 13

= 156

**(xii) 190969**

= √190969 = √19 × 19 × 23 × 13

= 19 × 23

= 437

**(xiii) 586756**

= √586756 = √2 × 2 × 383 × 383

= 2 × 383

= 766

**(xiv) 27225**

**∴** √27225 = √3 × 3 × 5 × 5 × 11 × 11

= 3 × 5 × 11

= 165

**(xv) 3013696**

= √3013696 = √2 × 2 × 2 × 2 × 2 × 2 × 217 × 217

= 2 × 2 × 2 × 217

= 1736

**Question no – (3)**

**Solution : **

Dear students, we can solve the problem as follows :

180 = (2 × 2) × 93 × 3) × 5

**∴** 5 is unpaired so 5 should be multiplied.

**∴** New required number = 2 × 2 × 3 × 5 × 5 = 900

√900 = √30 × 30 = 30

**Question no – (4)**

**Solution :**

147 = 3 × 7 × 7

**∴** 3 is unpaired so 3 should be multiplied.

**∴** New required number,

= 3 × 3 × 7 × 7

= 441

**∴** √441 = √21 × 21 = 21

**Question no – (5)**

**Solution :**

3645 = 3 × 3 × 3 × 3 × 3 × 3 × 5

**∴** 5 is unpaired so 5 should be divided.

**∴** New required number

= 3 × 3 × 3 × 3 × 3 × 3

= 729

**∴** √729 = √27 × 27

= 27

**Question no – (6)**

**Solution :**

1152 = (2 × 2) × (2 × 2) × (2 × 2) × 2 × (3 × 3)

**∴** 2 is unpaired so ‘2’ should be divided.

**∴** New required number,

= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3

= 576

∴ √576 = √24 × 24

= 24

**Question no – (7)**

**Solution :**

Let, One number = x

**∴** Other, 16x

According to question –

x × 16x = 1296

or, 16x² = 1296

or, x² = 1296/16 = 81

or, x = √81 = 9

**∴** One number = 9

**∴** Another number = 9 × 16 = 144

**Question no – (8)**

**Solution :**

Total collected many = 202500

**∴** Each paid as money rupees there were resident,

**∴** The number of resident = √202500

= √5 × 5 × 5 × 5 × 2 × 2 × 3 × 3 × 3

= 5 × 5 × 2 × 3 × 3

= 450

Therefore, the number of residents are 450.

**Question no – (9)**

**Solution :**

Total collected = 92.16 = 9216/100

**∴** member collected as many paise as there were members.

**∴** there no of member = √9216/100 = contribution of members

**∴** √9216/100

= √96 × 96/10 × 10

= 96/10

= 9.6 Paise.

**Question no – (10)**

**Solution :**

Total collected fees = 2304

The number of students,

= √2304

= √2 × 2 × 2 × 2 × 2 × 2 × 2 × 2× 3 × 3

= 2 × 2 × 2 × 2 × 3

= 48

Therefore, there are 48 students in the school.

**Question no – (11)**

**Solution :**

Given in the question,

Area of square = 5184 m²

**∴** One side of square,

= √5184 = √72 × 72

= 72

**∴** Perimeter of square,

= 4 × 72

= 288 m

Let, breadth of rectangle = x

Length of rectangle = 2x

According to question –

2 (x + 2x) = 288

or, 3x = 288/2 = 144

or, x = 144/3 = 48 m

**∴** Breadth = 48 m

Length = 2 × 48 = 96 m

**∴** Area of rectangular,

= 48 × 96

= 4608 m²

Thus, the area of the rectangular field is 4608 m².

**Question no – (12)**

**Solution :**

**(i)** L.C.M 6, 9, 15, 20 is = 180

180 = (2×2×3×3) × 5

To make perfect square, multiply it by 5.

**∴** Required number = (180 × 85) = 900

Thus, the least square number is 900.

**(ii)** Taking L.C.M, of 8, 12, 15, 20

L.C.M = 4×3×5×2 = 120

= (2×2×3×5×2)

To make it perfect square multiply it by 30

**∴** Required number,

= (120 × 30)

= 3600

Thus, the least square number is 3600.

**Question no – (13)**

**Solution :**

**(i) 121**

= 121 – 1 = 120

120 – 3 = 117

117 – 5 = 112

112 – 7 = 105

105 – 9 = 96

96 – 11 = 85

85 – 13 = 72

72 – 15 = 57

57 – 17 =40

40 – 19 = 21

21 – 21 = 0

**∴** There are 11 number subtracted.

**∴** √121 = 11

**(ii) 169**

= 169 – 1 = 168

168 – 3 = 165

165 – 5 = 160

160 – 7 = 153

153 – 9 = 144

144 – 11 = 133

133 – 13 = 120

120 – 15 = 120

120 – 15 = 105

105 – 17 = 88

88 – 21 = 48

48 – 23 = 25

25 – 25 = 0

**∴** There are 13 numbers subtracted.

**∴** √169 = 13

**Question no – (14)**

**Solution : **

**(i) 7744**

= √7744 = √2 × 2 × 2 × 2 × 2 × 2 × 11 × 11

= 2 × 2 × 2 × 11

= 88

**(ii) 9604**

= √9604 = √2 × 2 × 7 × 7 × 7 × 7

= 2 × 7 × 7

= 98

**(iii) 5929**

= √5929 = √7 × 7 × 11 × 11

= 7 × 11

= 77

**(iv) 7056**

= √7056 = √2 × 2 × 2 × 2 × 3 × 3 × 7 × 7

= 2 × 2 × 3 × 7

= 84

**Question no – (15)**

**Solution :**

Total student contributed = 2401

**∴** The number of student,

= √2401

= √7 × 7 × 7 × 7

= 7 × 7

= 49

Therefore, the number of students in the class is 49.

**Question no – (16)**

**Solution :**

Total student = 6000

Student left,

= 6000 – 71

= 5929

**∴** The number of row

= √5929

= √7 × 7 × 11 × 11

= 7 × 11

= 77

**∴** The number of rows are 77

**Squares and Square Roots Exercise 3.5 Solution : **

**Question no – (1) **

**Solution : **

**(i) 12544**

long division method :

**∴ **Square root of 12544 is 112.

**(ii) 97344**

long division method :

So, the square root of 97344 is = 312

**(iii) 286225**

long division method :

**∴** The square root of 286225 is = 535

**(v) 363609**

long division method :

So, the square root of 363609 is = 603

**(vi) 974169**

long division method :

So, the square root of 974169 is = 987

**(vii) 120409**

long division method :

**
∴ **The square root of 120409 is = 347

**(viii) 1471369**

long division method :

**
**Thus, the square root of 1471369 is = 1213

**(ix) 291600**

Long division method :

So, the square root of 291600 is = 540

**(x) 9653449**

Long division method :

So, the square root of 9653449 is = 3107

**(xi) 1745041
**

Long division method :

Thus, the square root of 1745041 is = 1321

**(xii) 4008004**

long division method :

∴ The square root of 4008004 is = 2002

**(xiii) 20657025
**

Long division method :

**
**So, the square root of 20657025 is = 4545

**(xiv) 152547201**

Long division method :

Thus, the square root of 152547201 is = 12351

**(xv) 20421361**

Long division method :

So, Square root of 20421361 is = 4519.

**(xvi) 62504836
**

Long division method :

**∴** The square root of 62504836 is = 7906.

**(xvii) 82264900**

Long division method :

Thus, the square root of 82264900 is = 9070

**(xviii) 3226694416**

Long division method :

So, the square root of 3226694416 is = 56804

**(xix) 6407522209**

Long division method :

∴ The square root of 6407522209 is = 80047

**(xx) 3915380329**

Long division method :

So, the square root of 3915380329 is = 62573

**Question no – (2) **

**Solution : **

**(i) 2361**

**∴** 57 should be subtracted

**(ii) 194491**

**∴** 10 should be subtracted

**(iii) 26535**

**∴** 291 should be subtracted

**(iv) 16160**

∴ 31 should be subtracted

**(v) 4401624**

**∴** 20 should be subtracted.

**Question no – (3) **

**Solution : **

**(i) 5607**

**∴** 131 should be added.

**(ii) 4931**

**∴** 110 should be added.

**(iii) 4515600**

**∴** 25 should be added.

**(iv) 37460**

**∴** 176 should be added.

**(v) 506900**

**∴** 44 should be added.

**Question no – (4) **

**Solution : **

The greatest number of 5 digit which a perfect square is 99856.

**Question no – (5) **

**Solution : **

The least number of 4 digits which a perfect square is 1024

**Question no – (6) **

**Solution : **

The least number of Six digits perfect square is 100489

**Question no – (7) **

**Solution : **

The greatest number of 4 digit perfect square 9801

**Question no – (8) **

**Solution : **

Total number = 8160

∴ Left = 60

∴ The number of soldier,

= 8160 – 60 = √8100

= √90 × 90

= 90

Therefore, the number of soldiers in each row is 90.

**Question no – (9) **

**Solution : **

Let, The One side of square = x

∴ Area one side of square = x²

According to question,

x² = 60025

or, x = √60025

= √245 × 245

= 245 m

∴ Perimeter of square,

= 245 × 4

= 980 m

= 0.98 km

A man cycles in 1 hr. 18 km

∴ A man cycles in 1 hr,

= 0.98/18

= 0.054 hr.

= 3 min 16 sec

Thus, in 3 min 16 sec he return at the starting point.

**Question no – (10) **

**Solution : **

Total cost of leveling and turfing a square = 13322.50

Total cost of leveling and turfing a square = 2.50/ per m²

∴ Total cost of leveling and turfing a square in 1 square m²

= 13222.50/2.50

= 5329 m²

Let, outside of square = x

∴ According to question –

x² = 5329

or, x = √5329 = 73 m

∴ Perimeter of square,

= 4 × 73

= 292 m

∴ Cost of perfect at 5 per metre,

= 292 × 5

= 1460

∴ The cost of fencing it at Rs 5 per metre is Rs. 1460.

**Question no – (11) **

**Solution : **

The greatest number of 3 digit of perfect square is 961

**Question no – (12) **

**Solution : **

Given number, 2300

∴ 2300 < (48)^{2}

∴ 4 should be added.

**Squares and Square Roots Exercise 3.6 Solution : **

**Question no – (1) **

**Solution : **

**(i) 441/961**

= √441/√961

= √21 × 21/√31 × 31

= 21/31

**(ii) 324/841**

= √324/√841

= √18 × 18/√29 × 29

= 18/29

**(iv) 2 14/25**

= √64/√25

= √8 × 8/√5 × 5

= 8/5

**(v) 2 137/196**

= √529/√196

= √23 × 23/√14 × 14

= 23/14

**(vi) 23 26/121**

= √2809/√121

= √53 × 53/√11× 11

= 53/11

**(vii) 25 444/729**

= √17311/√729

= √137 × 137/√27 × 27

= 137/27

**(viii) 75 46/49**

= √3721/√49

= √61 × 61/√7 × 7

= 61/7

**(ix) 3 942/2209**

= √7569/√2209

= √87 × 87/√47 × 47

= 87/47

**(x) 3 334/3025**

= √9409/√3025

= √97 × 97/√55 × 55

= 97/55

**(xi) 21 2797/3364**

= √73441/√3364

= √271 × 271/√58 × 58

= 271/58

**(xii) 38 11/25**

= √961/√25

= √31 × 31/√5 × 5

= 31/5

**(xiii) 23 394/729**

= √17161/729

= √131 × 131/√27 × 27

= 131/27

**(xiv) 21 51/169**

∴ √3600/169

= √60 × 60/√13 × 13

= 60/13

**(xv) 10 151/225**

**∴** √2401/225

= √49 × 49/√15 × 15

= 49/15

**Question no – (2) **

**Solution :**

**(i) √80/√405**

= √2 × 2 × 2 × 2 × 5/ √3 × 3 × 3 × 3 × 5

= 2 × 2 √5 / 3 × 3 √5

= 4/9

Therefore, the value will be 4/9.

**(ii) √441/ √625**

= √21 × 21/25 × 25

= 21/25

Therefore, the value will be 21/25

**(iii) √1587/1728**

= √3 × 23 × 23 /√2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

= 23 √3 /2 × 2 × 2 × 3√3

= 23/24

Hence, the value will be 23/24

**(iv) √72 × √338**

= √2×2×2×3×3 × √2×13×13

= 3 × 2 × 2 × 13

= 156

Thus, the value will be 156.

**(v) √45 × √20**

= √5×3×3 × √2×2×5

= 5 × 2 × 3

= 30

Thus, the value will be 30.

**Question no – (3)**

**Solution : **

Area of square,

= 80 224/179

= 58564/729 m²

**∴** Side of the square,

= √58564/729

= √242 × 242/√27× 27

= 242/27

= 8 26/27 m

Therefore, the length of each side of the field will be 8 26/27 m.

**Question no – (4)**

**Solution : **

Area of square,

= 30 1/4

= 121/4 m²

**∴** One side square,

= √121/4

= 11/2

= 5 1/2 m

Therefore, the length of the sides of the square will be 5 1/2 m.

**Question no – (5)**

**Solution : **

Area to rectangular field,

= 72 × 338

= 24336 m²

According to question

**∴** Area of rectangular,

= 24336 m² = Area of square field.

**∴** One side of square field,

= √24336

= √156 √ 126

= 156 m.

Thus, the length of side of square playground is 156 m.

**Squares and Square Roots Exercise 3.7 Solution : **

**Question no – (1)**

**Solution : **

Given number, 84.8241

= 848241/10000

= √921 × 921/ 100 × 100

= 921/100

= 9.21…(In decimal form)

**Question no – (2)**

**Solution : **

Given number, 0.7225

= 7225/10000

= √85 × 85 /100 × 100

= 85/100

= 0.85…(In decimal form)

**Question no – (3)**

**Solution : **

Given number, 0.813604

= 813604/1000000

= √902 × 902/1000 × 1000

= 902/10000

= 0.902…(In decimal form)

**Question no – (4)**

**Solution : **

Given number, 0.00002025

= 2025/100000000

= √45 × 45/10000 × 10000

= 45/10000

= 0.0045…(In decimal form)

**Question no – (5)**

**Solution : **

From the question, 150.0625

= 1500625/10000

= √1225 × 1225/100 × 100

= 1225/100

= 12.25…(In decimal form)

**Question no – (6)**

**Solution : **

Given number, 225.6004

= 2256004/10000

= √1502 × 1502/100 × 100

= 1502/100

= 15.02…(In decimal form)

∴ The square root of 225.6004 is 15.02

**Question no – (7)**

**Solution : **

Given number, 3600.720036

∴ 3600720036/100000

= √60006 × 60006/ 1000 × 1000

= 60006/1000

= 60.006…(In decimal form)

Hence, square root of 3600.720036 is 60.006

**Question no – (8)**

**Solution : **

From the question, 236.144689

= 236144689/1000000

= √15367 × 15367/ 1000 × 1000

= 15367/10000

= 15.367…(In decimal form)

Thus, the square root of 236.144689 is 15.367

**Question no – (9)**

**Solution : **

Given number, 0.00059049

= 59049/100000000

= √243 × 243/ 10000 × 10000

= 243/10000

= 0.0243…(In decimal form)

Thus, the square root of 0.00059049 is = 0.0243

**Question no – (10)**

**Solution : **

From the question, 176.252176

= 176252176/1000000

= √13276 × 13276/ 1000 × 1000

= 13276/1000

= 13.276…(In decimal form)

Hence, the square root of 176.252176 is 13.276

**Question no – (11)**

**Solution : **

Given number, 9998.0001

= 99980001/10000

= √9999 × 9999/ 100 × 100

= 9999/100

= 99.99…(In decimal form)

**∴** The square root of 9998.0001 is 99.99

**Question no – (12)**

**Solution : **

Given, 0.00038809

= 38809/100000000

= 197 × 197/ 10000 × 10000

= 197/10000

= 0.0197…(In decimal form)

Therefore, the square root of 0.00038809 is 0.0197

**Question no – (13)**

**Solution : **

From the question we get,

227.798649

= √227798649/1000000

= √15093 × 15093/ 1000 × 1000

= 15093/1000

= 15.093

Therefore, the required fraction will be 15.093

**Question no – (14)**

**Solution : **

A per the given question,

Area of square playground = 256.6406 m²

**∴** One side square playground,

= √256.6404

= √2566404/10000

= √1602 × 1602/ 100×100

= 16.02 m

Hence, the length of one side of the playground will be 16.02 m.

**Question no – (15)**

**Solution : **

Given, 0.00053361

**∴** √53361/ 100000000

= √231 × 231/ 10000 × 10000

= 231/10000

= 0.0231

Thus, the required fraction will be 0.0231

**Question no – (16)**

**Solution : **

**(i) √59.29 – √5.29/√59.29 + √5.29**

= (√59.29 – 5.29) (59.29 – √5.29)/(59.29 + √5.29) (√59.29 – √5.29)

= (√59.29 – √5.29)/(59.29) – (√5.29)²

= (59.29) – 2 × √59.29 √5.29 + (√5.29)²/ 59.29 – 5.29

= 59.29 + 5.29 – 2 × (√5929/ 100 × 529/100)/54

= 64.58- (2 × √77 × 77/ 10 × 10 × 23 × 23/10 × 10/54)

= 64.58 – (2 × 7 × 23/10 × 10)/54

= 64.58 – 3542/100/54 = 64.58 – 35.42/54

= 29.16/54

= 0.54…(Simplified)

**(ii) √0.2304 + √0.1764/√0.2304 – √0.1764**

= 0.2304 + √0.1764/ 0.2304 – √0.1764

= (√0.2304 + √0.1764) (√0.2304 + √0.1764)/(√0.2304 – √0.1764) (√0.2304 + √0.1764)

= (√0.2304 + (√0.1764)²/ (√0.2304)² – (√0.1764)²

= (√0.2304)² + 2 × √0.2304 × √0.1764 + (√0.1764)²/ 0.2304 – 0.1764

= (0.2304 + 0.1764) – 2 × √2304/10000 × 1764/1000/ 0.54

= (0.4068 – 2√48×48/ 100 × 100)/0.054

= 0.4068 – (2 × 48/100 ×48/100)

= 0.4068- (4032/10000)/0.054

= 0.4068 – 0.4032/0.054

= 0.0036/0.054

= 0.0060…(Simplified)

**Question no – (17)**

**Solution : **

Evaluate √50625

= √225 × 225

= 225

Now, the value of √506.25 + √5.0625

= √50625/100 + √50625/10000

= 225/10 + 225/100

= 225 + 225/100

= 2475/100

= 24.75

Therefore, the value is 24.75

**Question no – (18)**

**Solution :**

The value of, √130.0225

= √103.0225/√10000

= √1015 × 1015/√100 × 100

= 1015/100

= 10.15

The value of, (i) √10302.25

= √10302.25/100

= √1015 × 1015/10 × 10

= 1015/10

= 101.5

Thus, the value will be 101.5

The value of, (ii) √1.030225

= √1.030225/1000000

= √1015 × 1015/1000 × 1000

= 1015/1000

= 1.015

Therefore, the value will be 1.015

**Squares and Square Roots Exercise 3.8 Solution : **

**Question no – (1)**

**Solution : **

**(i) Given number, 5**

**∴** √5 = 2.236

So, square root of 5 correct to three places of decimal is 2.236.

**(ii) Given number, 7**

**∴** √7 = 2.646

So, the square root of 7 correct to three places of decimal is 2.646

**(iii) Given number, 17**

**∴** √17 = 4.123

So, the square root of 17 correct to three places of decimal is 4.123

**(iv) Given number, 20**

**∴** √20 = 4.472

Hence, the square root of 20 correct to three places of decimal is 4.472

**(v) Given number, 66**

∴ √66 = 8.1240

Thus, the square root of 66 correct to three places of decimal is 8.1240

**(vi) We have, 427**

∴ √427 = 20.664

Hence, the square root of 427 correct to three places of decimal is 20.664

**(vii) We have, 1.7**

∴ √1.7 = 1.304

Therefore, the square root of 1.7 correct to three places of decimal is 1.304

**(viii) Given number, 23.1**

∴ √23.1= 4.602

Thus, the square root of 23.1 correct to three places of decimal is 4.602

**(ix) Given number, 2.5**

∴ √2.5 = 1.5811

∴ The square root of 2.5 correct to three places of decimal is 1.5811

**(x) Here we have, 237.615**

∴ √237.615 = 15.415

So, the square root of 237.615 correct to three places of decimal is 15.415

**(xi) We have, 15.3215**

∴ √15.3215 = 3.914

So, the square root of 15.3215 correct to three places of decimal is 3.914

**(xii) Given number, 0.9**

∴ √0.9 = 0.949

So, the square root of 0.9 correct to three places of decimal is 0.949

**(xiii) Here we have, 0.1**

∴ √0.1 = 0.316

Hence, the square root of 0.1 correct to three places of decimal is 0.316

**(xiv) Given number, 0.016**

∴ √0.16 = 0.126

∴ The square root of 0.16 correct to three places of decimal is 0.126

**(xv) Given number, 0.00064**

∴ √0.00064 = 0.025

Hence, the square root of 0.00064 correct to three places of decimal is 0.025

**(xvi) Here we have, 0.019**

∴ √0.019 = 1.138

So, the square root of 0.019 correct to three places of decimal is 1.138

**(xvii) Given number, 5/12**

= 0.41666666

∴ √5/12 = 0.645

Hence, the square root of 5/12 correct to three places of decimal is 0.645

**(xviii) Given in the question, 12.0068**

∴ √12.0068 = 3.4651

So, the square root of 12.0068 correct to four decimal places 3.4651.

**Question no – (2)**

**Solution :**

Given in the question, 12.0068

**∴** √12.0068 = 3.4651

Therefore, the square root of 12.0068 correct to four decimal places 3.4651.

**Question no – (3)**

**Solution : **

Given in the question, 11

∴ √11 = 3.31662

Hence, the square root of 11 correct to five decimal places is 3.31662

**Question no – (4)**

**Solution : **

**(i) √144/7**

= √11 × 11 /√7

= 11/√7

= √11 √ 7/ √7 × √7

= 11√7/7

= 11 × 2.646/7

= 4.536

**(ii) √23500/3**

= √2500/√3

= √50 × 50/√3

= 50/√3 = 50√/√3 × √3

= 50√3/3

= 50 × 1.732/3

= 28.867

**Question no – (5)**

**Solution : **

**(i) 196/75**

√196/√75 = √2 × 2 × 7 × 7/ 3 × 5 × 5

= 2 × 7/ 5 √3

= 14/ 5 × 1.732

= 1.617

**(ii) 400/63**

×400/63 = √2 × 2 × 2 × 2 × 5 × 5/ √2 × 3 × 3 × 3

= 2 × 2 × 5/ 3 × √2 × √3

= 20/ 3 × 1.412 × 1.732

= 2.520

**(iii) 150/7**

√150/√7 = √3 × 2 × 5 × 5/ √7

= √3 × √2 × 5 /√7

= 1.412 × 1.732 × 5/2.646

= 4.628

**(iv) 256/5**

√256/√5 = √2 × 2 × 2 × 2× 2 × 2 × 2 × 2/√5

= 2 × 2 × 2 × 2/2.236

= 8/2.236

= 7.155

**(v) 27/50**

√27/50 = √3 × 3 × 3 / √2 × 5 × 5

= 3 √3/5 √2

= 3 × 1.732/ 5 × 1.414

= 0.735

**Squares and Square Roots Exercise 3.9 Solution : **

**Question no – (1)**

**Solution : **

Given, 7

√7 = 2.646

**Question no – (2)**

**Solution : **

Given, 15

√15 = 3.873

**Question no – (3****)**

**Solution : **

Given, 74

√74 = 8.602

**Question no – (4)**

**Solution : **

Given, 82

√82 = 9.055

**Question no – (5)**

**Solution : **

Given, 198

√198 = √2 × 3 × 11

= 1.414 × 1.732 × 3.317

= 14.070

**Question no – (6)**

**Solution : **

Given, 540

√540 = √2 × 5 × 3 × 3 × 3 × 2

= 2 × 3 √15 = 6 × 3.873

= 23.24

**Question no – (7)**

**Solution : **

Given, 8700

√8700 = √87 × 100 × 10

= 10 √87 = 10 × 9.327

= 9327

**Question no – (8)**

**Solution : **

Given, √3509

√3500 = √11 × 11 × 29

= 11 √29 = 11 × 5.385

= 59.235

**Question no – (9)**

**Solution : **

Given, 6929

√6929 = √13 × 13 × 41 = 13 √41

= 13 × 6.40

= 83.239

**Question no – (10)**

**Solution : **

Given, 25725

√25725 = √3 × 5 × 5 × 7 × 7 × 7

= 5 × 7 √3 × 7

= 35 √21

= 35 × 4.583

= 157.78

**Question no – (11)**

**Solution : **

Given, 1312

√13123 = √2 × 2 × 2 × 2 × 2 × 41 = 2 × 2 √82

= 4 × 9.055 = 36.22

**Question no – (12)**

**Solution : **

Given, 4192

√4192 = √2 × 2 × 2 × 2 × 131

**Question no – (13)**

**Solution : **

Given, 4955

√5955 = √5 × 99

**Question no – (14)**

**Solution : **

Given, 99/144

√99/144 = √3 × 3 × 11/12 × 12

= 3/12 √11

= 1/4 × 3.317

= 0.829

**Question no – (15)**

**Solution : **

Given, 57/169

√57/169 = √57/13 × 13 √57/13

= 7.550/13

= 0.580

**Question no – (16)**

**Solution : **

Given, 101/169

√101/169 = √101/13 × 13

= √101/13

**Question no – (17)**

**Solution : **

Given, 13.21

√13.12 = √1321/100

= 1321/10

**Question no – (18)**

**Solution : **

Given, 21.97

√21.97 = √2197/√100

= √13 × 13 × 13/√10 × 10

= 13/10 √13 = 1.3 × 3 .606

= 4.6878

**Question no – (19)**

**Solution : **

Given, 110

√110 = √11 × 5 × 2

= 3.167 × 2.236 × 1.414

= 10.013

**Question no – (20)**

**Solution : **

Given, 1110

√1110

= √111 × 2 × 5

**Question no – (21)**

**Solution : **

Given, 11.11

√11.11

= √1111/√100

= √1111/10

**Question no – (22)**

**Solution :**

As per the question,

Area = 325 m²

**∴** one side length,

= √325 = √5 × 5 × 13

= 5 √13

= 5 × 3.606

= 18.03 m

Therefore, the approximate length of one side of the field will be 18.03 m.

**Question no – (23)**

**Solution :**

Area of rectangle = Area of square

= 240 × 70

= 16800 m²

**∴** One side of square,

= √16800 = √2 × 2 × 2 × 2 × 2 × 3 × 7 × 5 × 5

= 2 × 2 × 5 = √2 × 3 × 7

= 20 √42 = 20 × 6.481

= 129.62 m

Thus, the length of the side of a square will be 129.62 m.

**Next Chapter Solution : **

👉 Chapter 4 👈