# Chapter – 6: Squares and Square Roots

# 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 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.

**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

**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. 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

**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. 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}

**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

**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

**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.

# 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} + 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

**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.

## Exercise 6.3

**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.

**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

**4. Find the square roots of the following numbers by the Prime Factorization 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 = 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

iv.

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.

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.

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 = 7×7×11×11

⇒ 5929 = (7×7)×(11×11)

⇒ 5929 = (7×11)×(7×11)

⇒ 5929 = (7×11)^{2}

⇒ √5929 = 7×11 = 77

viii.

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 = 23×23

529 = (23)^{2}

√529 = 23

x.

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.

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

1764 = 2×2×3×3×7×7

⇒ 1764 = (2×2)×(3×3)×(7×7)

⇒ 1764 = 2^{2}×3^{2}×7^{2}

⇒ 1764 = (2×3×7)^{2}

⇒ √1764 = 2×3×7 = 42

ii.

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 = 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

7056 = 2×2×2×2×3×3×7×7

⇒ 7056 = (2×2)×(2×2)×(3×3)×(7×7)

⇒ 7056 = 2^{2}×2^{2}×3^{2}×7^{2}

⇒ 7056 = (2×2×3×7)^{2}

⇒ √7056 = 2×2×3×7 = 84

iv.

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

6084 = 2×2×3×3×13×13

⇒ 6084 = (2×2)×(3×3)×(13×13)

⇒ 6084 = 2^{2}×3^{2}×13^{2}

⇒ 6084 = (2×3×13)^{2}

⇒ √6084 = 2×3×13 = 78

v.

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

2916 = 2×2×3×3×3×3×3×3

⇒ 2916 = (3×3)×(3×3)×(3×3)×(2×2)

⇒ 2916 = 3^{2}×3^{2}×3^{2}×2^{2}

⇒ 2916 = (3×3×3×2)^{2}

⇒ √2916 = 3×3×3×2 = 54

vi.

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

2304 = 2×2×2×2×2×2×2×2×3×3

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

⇒ 2304 = 2^{2}×2^{2}×2^{2}×2^{2}×3^{2}

⇒ 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.

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

36 = 2×2×3×3

⇒ 36 = (2×2)×(3×3)

⇒ 36 = 2^{2}×3^{2}

⇒ 36 = (2×3)^{2}

⇒ √36 = 2×3 = 6

ii.

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 = 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 = 5×23×23

⇒ 2645 = (23×23)×5

Here, 5 cannot be paired.

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

New number = 2645 ÷ 5 = 529

529 = 23×23

⇒ 529 = (23)^{2}

⇒ √529 = 23

v.

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

400 = 2×2×2×2×5×5

⇒ 400 = (2×2)×(2×2)×(5×5)

⇒ 400 = (2×2×5)^{2}

⇒ √400 = 20

vi.

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

**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=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

**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 =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.

**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 (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:

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

**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.

∴ √2304 = 48

ii.

∴ √4489 = 67

iii.

∴ √3481 = 59

iv.

∴ √529 = 23

v.

∴ √3249 = 57

vi.

∴ √1369 = 37

vii.

∴ √5776 = 76

viii.

∴ √7921 = 89

ix.

∴ √576 = 24

x.

∴ √1024 = 32

xi.

∴ √3136 = 56

xii.

∴ √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.

∴ √144 = 12

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

ii.

∴ √4489 = 67

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

iii.

√27225 = 165

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

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.**

∴ √2.56 = 1.6

ii.

∴ √7.29 = 2.7

iii.

∴ √51.84 = 7.2

iv.

∴ √42.25 = 6.5

(v)

∴ √31.36 = 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.

∴ √400 = 20

∴ We must subtract

2 from 402 to get a perfect square.

New number = 402 – 2 = 400

∴ √400 = 20

ii.

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

∴ √1936 = 44

iii.

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

New number = 3250 – 1 = 3249

∴ √3249 = 57

iv.

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

New number = 825 – 41 = 784

∴ √784 = 28

∴ We must subtract 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)

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

∴ √529 = 23

(ii)

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)

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

∴ √256 = 16

(iv)

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

∴ √1849 = 43

(v)

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

**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

∴ 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.

Given, AB = 6 cm, BC = 8 cm

Let AC be x cm.

∴ AC2 = AB2 + BC2

Hence, AC = 10 cm.

b.

Given, AC = 13 cm, BC = 5 cm

Let AB be x cm.

∴ AC2 = AB2 + BC2

⇒ AC2 – BC2 = AB2

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

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

Hence, 16 children would be left out in the arrangement

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