**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)**One’s digit in the square root of 9801 is equal to 1 or 9.

[9^{2}=81 whose unit place is 1].

**(ii)** One’s digit in the square root of 99856 is equal to 6.

[6^{2}=36 and 4^{2}=16, both the squares have unit digit 6].

**(iii) **One’s digit in the square root of 998001 is equal to 1 or 9.

[9^{2}=81 whose unit place is 1].

**(iv)** One’s digit in the square root of 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 is not perfect square because it ends with 3.

**ii. **257 is not perfect square because it ends with 7

**iii.** 408 is not perfect square because it ends with 8

**iv.** 441 is perfect square because it ends with 1

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

**Solution:**

According to question:

Square root of 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.

Hence, √100 = 10

Now, square root of 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.

Hence, √169 = 13

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

i. 729

ii. 400

iii. 1764

iv. 4096

v. 7744

vi. 9604

vii. 5929

viii. 9216

ix. 529

x. 8100

**Solution:**

**i.**

729 = 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 = 2 × 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.

So, 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 = 22×32×72

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.

So, 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 = 22×32×52

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.

So, 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 = 22×22×32×72

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.

So, 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 = 22×32×132

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.

So, 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 = 32×32×32×22

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.

So, 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 = 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.

252 = 2×2×3×3×7

= (2×2)×(3×3)×7

Here, 7 cannot be paired.

So, 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.

So, 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.

So, 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.

So, 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.

So, 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.

So, 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 us assume the number of students in the school be, x.

Each student donate Rs.x .

Total rupees 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

Hence, 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 us assume the number of rows be, x.

Total number of plants in each rows = x.

Total rupees contributed by all the students = x × x = x^{2}

Given,

X^{2} = Rs.2025

x2 = 3×3×3×3×5×5

X^{2} = (3×3)×(3×3)×(5×5)

X^{2} = (3×3×5)×(3×3×5)

X^{2} = 45×45

x = √45×45

x = 45

Hence, The number of rows = 45

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 = (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 = (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) to get perfect square.

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

= 3600

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