The document NCERT Solutions(Part- 1)- Squares and Square Roots Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.

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**Question 1.**** Find the perfect square numbers between (i) 30 and 40 and (ii) 50 and 60.**

**Solution: **

**(i) **Since, 1 *1= 1 2 * 2= 4 3 * 3= 9

4 * 4= 16 5 * 5= 25 6 * 6= 36

7 * 7= 49

Thus, 36 is a perfect square number between 30 and 40.

**(ii)** Since, 7 * 7 = 49 and 8 * 8 = 64. It mean there is no perfect number between 49 and 64; and thus there is not perfect number between 50 and 60.

**Question 2. ****Can we say whether the following numbers are perfect squares? How do we know?**

**(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 1069 (vi) 2061**

**Write five numbers which you can decide by looking at their one’s digit that they are not square numbers.**

**Solution: **

**(i)** 1057

∵ The ending digit is 7 (which is not one of 0, 1, 4, 5, 6 or 9)

∴ 1057 cannot be a square number.

**(ii)** 23453

∵ The ending digit is 3 (which is not one of 0, 1, 4, 5, 6 and 9).

∴ 23453 cannot be a square number.

**(iii)** 7928

∵ The ending digit is 8 (which is not one of 0, 1, 4, 5, 6 and 9).

∴ 7928 cannot be a square number.

**(iv)** 222222

∵ The ending digit is 2 (which is not one of 0, 1, 4, 5, 6 or 9).

∴ 222222 cannot be a square number.

**(v) **1069

∵ The ending digit is 9.

∴ It may or may not be a sqaure number.

Also,

30 * 30 = 900

31 * 31 = 961

32 * 32 = 1024

33 * 33 = 1089

i.e. No natural number between 1024 and 1089 is a square number.

∴ 1069 cannot be a square number.

**(vi)** 2061

∵ The ending digit is 1

∴ It may or may not be a square number.

∵ 45 * 45 = 2025

and 46 * 46 = 2116

i.e. No natural number between 2025 and 2116 is a square number.

∴ 2061 is not a square number.

We can write many numbers which do not end with 0, 1, 4, 5, 6 or 9. (i.e. which are not square number). Five such numbers can be:

1234, 4312, 5678, 87543, 1002007.

**Question 3.** **Write five numbers which you cannot decide just by looking at their unit’s digit (or one’s place) whether they are square numbers or not.**

**Solution:** Any natural number ending in 0, 1, 4, 5, 6 or 9 can be or cannot be a square-number. Five such numbers are:

56790, 3671, 2454, 76555, 69209

**Property 2. **If a number has 1 or 9 in the unit’s place, then its square ends in 1.

For example: (1)^{2 }= 1, (9)^{2} = 81, (11)^{2} = 121, (19)^{2} = 361, (21)^{2} = 441.

**Question 4: **Which of 1232, 772, 822, 1612, 1092 would end with digit 1?

**Solution:** The squares of those numbers end in 1 which end in either 1 or 9.

∴ The squares of 161 and 109 would end in 1.

**Property 3. **When a square number ends in 6, then the number whose square it is, will have 4 or 6 in its unit place.

**Question 5:** **Which of the following numbers would have digit 6 at unit place.**

**(i) 19 ^{2 }(ii) 24^{2} (iii) 26^{2} (iv) 36^{2} (v) 34**

**Solution: **(i)19^{2}: Unit’s place digit = 9

∴ 19^{2 }would not have unit’s digit as 6.

**(ii)** 24^{2}: Unit’s place digit = 4

∴ 24^{2} would have unit’s digit as 6.

**(iii)** 26^{2}: Unit’s place digit = 6

∴ 26^{2} would have 6 as unit’s place.

**(iv)** 36^{2}: Unit place digit = 6

∴ 36^{2} would end in 6.

**(v) **34^{2}: Since, the unit place digit is 4

∴ 34^{2} would have unit place digit as 6.

**Question 6: ****What will be the “one’s digit” in the square of the following numbers?**

**(i) 1234 (ii) 26387 (iii) 52698 (iv) 99880 (v) 21222 (vi) 9106**

**Solution: **

**(i) **∵ Ending digit = 4 and 4^{2} = 16

∴ (1234)^{2} will have 6 as the one’s digit.

**(ii)** ∵ Ending digit is 7 and 7^{2 }= 49

∴ (26387)^{2} will have 9 as the one’s digit.

**(iii) **∵ Ending digit is 8, and 8^{2} = 64

∴ (52692)^{2} will end in 4.

**(iv) **∵ Ending digit is 0.

∴ (99880)^{2} will end in 0.

**(v)** ∵ 2^{2} = 4

∴ Ending digit of (21222)^{2} is 4.

**(vi) **∵ 6^{2} = 36

∴ Ending digit of (9106)^{2 }is 6.

**Property 4.** A square number can only have even number of zeros at the end.

**Property 5. **The squares of odd numbers are odd and the squares of even numbers are even.

**Question 6.**** The square of which of the following numbers would be an odd number/an even number? Why?**

**(i) 727 (ii) 158 (iii) 269 (iv) 1980**

**Solution: **

**(i) **727

Since 727 is an odd number.

∴ It square is also an odd number.

**(ii)** 158

Since 158 is an even number.

∴ Its square is also an even number.

**(iii)** 269

Since 269 is an odd number.

∴ Its square is also an odd number

**(iv)** 1980

Since 1980 is an even number.

∴ Its square is also an even number.

**Question 7.**** What will be the number of zeros in the square of the following numbers?**

**(i) 60 (ii) 400**

**Solution: **

**(i) **In 60, number of zero is 1

∴ Its square will have 2 zeros.

**(ii) **∵ There are 2 zeros in 400.

∴ Its square will have 4 zeros.

**Property 6. **The difference between the squares of two consecutive natural numbers is equal to the sum of the two numbers.

**Property 7.** There are 2n non-perfect square numbers between the squares of the numbers n and n + 1.

**Question 8. ****How many natural numbers lie between 9 ^{2} and 10^{2}? Between 11^{2} and 12^{2}?**

**Solution: **(a) Between 9^{2} and 10^{2}

Here, n = 9 and n + 1 = 10

∴ Natural number between 9^{2} and 10^{2} are (2 * n) or 2 * 9, i.e. 18.

(b) Between 11^{2} and 12^{2}

Here, n = 11 and n + 1 = 12

∴ Natural numbers between 11^{2 }and 12^{2} are (2 * n) or 2 *11, i.e. 22.

**Question 9.**** How many non-square numbers lie between the following pairs of numbers:**

**(i) 100 ^{2 }and 101^{2} (ii) 90^{2} and 91^{2} (iii) 1000^{2} and 1001**

Solution: **(i)** Between 100^{2} and 101^{2}

Here, n = 100

∴ n * 2 = 100 * 2 = 200

∴ 200 non-square numbers lie between 1002 and 1012.

**(ii)** Between 90^{2} and 91^{2}

Here, n = 90

∴ 2 * n = 2 * 90 or 180

∴ 180 non-square numbers lie between 90 and 91.

**(iii) **Between 1000^{2} and 1001^{2}

Here, n = 1000

∴ 2 * n = 2 * 1000 or 2000

∴ 2000 non-square numbers lie between 1000^{2} and 1001^{2}.

**Property 8. **The sum of first n odd natural numbers is n^{2}.

or

If there is a square number, it has to be the sum of the successive odd numbers starting from 1.

**Question 10:**** Find whether each of the following numbers is a perfect square or not?**

**(i) 121 (ii) 55 (iii) 81 (iv) 49 (v) 69**

**Solution:**

**Remember**

If a natural number cannot be expressed as a sum of successive odd natural numbers starting from 1, then it is not a perfect square.

**(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

i.e. 121 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21. Thus, 121 is a perfect square.

**(ii) **55

∵ 55 – 1 = 54 30 – 11 = 19

54 – 3 = 51 19 – 13 = 6

51 – 5 = 46 6 – 15 = –9

46 – 7 = 39

39 – 9 = 30

Since, 55 cannot be expressed as the sum of successive odd numbers starting from 1.

∴ 55 is not a perfect square.

**(iii)** 81

Since,

81 – 1 = 80

80 – 3 = 77

77 – 5 = 72

72 – 7 = 65

65 – 9 = 56

56 – 11 = 45

45 – 13 = 32

32 – 15 = 17

17 – 17 = 0

∴ 81 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17.

Thus, 81 is a perfect square.

**(iv)** 49

Since, 49 – 1 = 48

48 – 3 = 45

45 – 5 = 40

40 – 7 = 33

33 – 9 = 24

24 – 11 = 13

13 – 13 = 0

∴ 49 = 1 + 3 + 5 + 7 + 9 + 11 + 13

Thus, 49 is a perfect square.

**(v) **69

Since

69 – 1 = 68

68 – 3 = 65

65 – 5 = 60

60 – 7 = 53

53 – 9 = 44

44 – 11 = 33

33 – 13 = 20

20 – 15 = 5

5 – 17 = –12

∴ 69 cannot be expressed as the sum of consecutive odd numbers starting from 1.

Thus, 69 is not a perfect square.

**Property 9. **The square of an odd number can be expressed as the sum of two consecutive natural numbers.

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