March 27, 2025

NCERT Class 8 Maths chapter square and square roots | Exemplar Solution

Class 8 Maths Chapter Square and Square Roots (Exemplar Solution)

Understanding squares and square roots is an important part of learning mathematics in Class 8. These concepts help build a strong foundation for more advanced math topics you’ll encounter later. The NCERT Class 8 Maths Exemplar provides various problems to help students practice and understand these concepts better. In this blog, we have simple and clear solutions to these problems, making it easier for you to learn and solve them.

In each of the questions, 1 to 24, write the correct answer from the given four options.
1. 196 is the square of
(a) 11                    (b) 12             (c) 14                (d) 16
Ans. (c)
Since  equation , this implies 196 is square of 14.

2. Which of the following is a square of an even number?
(a) 144                 (b) 169          (c) 441            (d) 625
Ans.(c)
Square of an even number ends with an even number. So 144 is a square of an even number.
12 x 12 =144.

3. A number ending in 9 will have the units place of its square as
(a) 3                     (b) 9              (c) 1                  (d) 6
Ans. (c)  Example : 9x 9 =81 , 19 x 19= 361 etc.

4. Which of the following will have 4 at the units place?
(a) equation

                (b) equation           (c) equation             (d) equation
Ans. (b)  Since class 8 maths chapter square and square roots

5. How many natural numbers lie between equation and equation?
(a) 9                    (b) 10           (c) 11               (d) 12
Ans. (b)Since there are ‘2n’ natural numbers between n and ( n+1). So there are 2 x 5=10 natural numbers between equation and equation.

6. Which of the following cannot be a perfect square?
(a) 841               (b) 529        (c) 198             (d) All of the above
Ans. (c)  A number ending with digits 2, 3, 7 or 8 can never be a perfect square. So, 198 cannot be written in the form of a perfect square.

7. The one’s digit of the cube of 23 is
(a) 6                   (b) 7             (c) 3                   (d) 9
Ans.  (b) The cubes of the numbers ending with digits 3  have 7 at one’s digit.
So, the one’s digit of the cube of 23 is 7. Also  class 8 maths chapter square and square roots .

8. A square board has an area of 144 square units. How long is each side of the board?
(a) 11 units         (b) 12 units               (c) 13 units          (d) 14 units
Ans. (b) Let the side of the square board be “equation” units. Then the area of the square board = equationunits.
Given that area of square board = 144 units
equation  units
equation units
equation units
Hence, the length of each side of board is 12units.

9. Which letter best represents the location of equation on the number line?
(a) A           (b)B         (c)C       (d) D
Ans. (c) C
Since equation
Therefore, the point “C” on the number line represents  equation .

10. If one member of a Pythagorean triplet is 2equation, then the other two members are
(a) equation            (b) equation          (c) equation     (d) equation
Ans.  (b)equation
For every number equation , 2equation , equation  form a  Pythagorean triplet.

11.  The sum of successive odd numbers 1, 3, 5, 7, 9, 11, 13 and 15 is
(a) 61   (b)   64   (c) 49   (d) 36
Ans. (b) We know that, the sum of first  equation odd natural numbers is  equation.
Given odd numbers are 1,3, 5, 7, 9,11,13 and 15.
So,  total number of  given odd numbers, equation.
Hence the sum of given odd numbers = equation.

12. The sum of first n odd natural numbers is
(a) equation            (b)  equation            (c)  equation                 (d) equation
Ans.  (b) Consider the following
1 [one odd number] = 1 = equation
1 + 3 [sum of first two odd numbers] = 4 = equation
1 + 3 + 5 [sum of first three odd numbers] = 9 = equation
1 + 3 + 5 + 7 [sum of first four odd numbers ] = 16 = equation
1 + 3 + 5 + 7 + 9 [sum of first five odd numbers ] = 25 = equation
1 + 3 + 5 + 7 + 9 + 11 [sum of first six odd numbers ] = 36 = equation
So we can say that the sum of first n odd natural numbers is equation.

13. Which of the following numbers is a perfect cube?
(a) 243   (b) 216   (c) 392 (d) 8640
Ans. (b)
216 = 2  x 2 x  2 x 3 x 3 x 3
Grouping the factors in triplets of equal factors, we get 216 = (2 x 2 x 2) x (3 x 3 x 3)
Clearly, in grouping, the factors of triplets of equal factors, no factor is left over.
So, 216 is a perfect cube.

14. The hypotenuse of a right angled triangle with its legs of lengths 3equation x  4equationis
(a) 5equation                                 (b )7equation                               (c) 16equation                            (d) 25equation
Ans.  (a)
Lengths of the  legs of the right angled triangle is 3equation and 4equation.
Using Pythagoras Theorem, Hypotenuse = equation
=equation
=equation
=equation

15.  The next two numbers in the number pattern 1, 4, 9,16, 25,… are
(a) 35, 48                          (b) 36, 49                (c) 36, 48              (d) 35, 49
Ans.(b)
We have, 1,4, 9,16, 25, ….
First number of the sequence is = 1= equation
Second number of the sequence is = 4=equation
Third number of the sequence is = 9=equation
Forth  number of the sequence is =16 =equation
Fifth number of the sequence is = 25=equation
Hence, the next two numbers are equation and equation, i.e. 36 and 49.

16.Which among equationequation would end with digit 1?
(a)  equation         (b)equation         (c)equation          (d)equation
Ans. (d)
Since equation
equation
equation
equation
Hence square of 59 would end with digit 1.

17.  A perfect square can never have the following digit in its one’s place.
(a) 1                                  (b) 8                             (c) 0                                    (d) 6
Ans. (b)
A number ending with digits 2, 3, 7 or 8 can never be a perfect square.  So , a perfect square can never have the digit 8 in its one’s place.

18. Which of the following numbers is not a perfect cube?
(a) 216                   (b) 567                   (c) 125              (d) 343
Ans.(b)
Prime factorization of 216, 567, 125 and 343 are
216=6 x 6 x 6,
567 = 3 x 3 x 3 x 3 x 7
125 = 5 x 5 x 5,
343 = 7 x 7 x 7
Clearly, 567 is not a perfect cube, because  grouping  in triplets of equal factors, we are left with two factors 3 x 7.

19. equation  is equal to
(a)10        (b)100          (c)1        (d)none of these
Ans.  (a) we know 1000=10 x 10 x 10 .
Taking cube root of both the sides
equation = class 8 maths chapter square and square roots
equation=10
Hence cube root 1000 is 10 .

20. If  equation is the square of a natural number equation , then equation is
(a)  the square of equation              (b) greater than equation                 (c) equal to equation            (d) equation
Ans. (d)
Given that equation.
Taking square root of both the side, we get equation = equation

21. A perfect square number having  equation digits, where equation is even, will have square root with
(a) equation digit                   (b) equation  digit            (c) equation  digit        (d) equation
Ans. (b) equation  digit

22. If equation is the cube root of equation, then  equation is
(a) equation          (b) equation                   (c)equation            (d) equation
Ans. (a)

23.  The value of class 8 maths chapter square and square roots  is
(a) 14         (b) 12                (c) 16            (d)13
Ans. (c)
equation = equation
= equation
= equation
= equation
=16

24. Given that equation , the value of equation is
(a) 74               (b) 60.4              (c) 64.4                  (d) 70.4
Ans. (d) 70.4
equation
= equation
= equation
=6.4
Hence  equation  = 64+6.4
=70.4

In questions 25 to 48, fill in the blanks to make the statements true.
25.  There are________perfect squares between 1 and 100.
An.  8
There are 8 perfect squares between 1 and 100, i.e. 4, 9,16, 25, 36, 49, 64 and 81.

26.  There are________ perfect cubes between 1 and 1000.
Ans.8
There are 8 perfect cubes between 1 and 1000, i.e. 8, 27,64,125, 216, 343 and 729.

27.  The unit’s digit in the square of 1294 is________
Ans. 6
We know that, the unit’s digit of the square of a number having digit  4 at unit’s place  is 6.
Hence, the units digit in the square of 1294 is 6.

28.  The square of 500 will have   ……… zeroes.
Ans.  4
Since equation
Hence, the square of 500 will have 4 zeroes.

29. There are  ………………….. natural numbers between equation  and equation.
Ans. equation
Between equation(=1) and  equation there are two (i.e., 2 × 1) non square numbers 2, 3.
Between equation  and equation there are four (i.e., 2 × 2) non square numbers 5, 6, 7, 8.
Natural number between equation  and equation =equation
=equation
=equation
Hence there are equation natural numbers between the square of equation and equation .

30.The square root of 24025 will have_______digits.
Ans. 3
If number of digits  equation in any given number  is odd , then number of digits in the square root of that number= equation
Here equation =5
Hence number of digits in the square root of 24025 = equation
=equation
=3

31. The square of 5.5 is________
Ans.  30.25
Square of 5.5= equation
= 30.25

32. The square root of 5.3 x 5.3 is________
Ans. 5.3
Square root of 5.3 x 5.3 = equation
=equation
=5.3

33. The cube of 100 will have________zeroes.
Ans.  6
Cube of 100 = equation
= 1000000

34. equation
Ans.  equation
equation
=equation
=equation

35.  equation
Ans.  equation
equation
=equation

36. One’s digit in the cube of 38 is________
Ans. 2
Since equation
=54872

37. The square of 0.7 is________
Ans: 0.49
Square of 0.7 = equation
=0.49

38.  The sum of first six odd natural numbers is________
Ans. 36
Since sum of first ‘equation‘ odd natural numbers = equation
So, sum of first ‘6’ odd natural numbers = equation
=36

39. The digit at the one’s place of  equation is________
Ans. 9
Since the unit’s digit of a square of a number  having digit at unit’s place as 3 or 7 is 9.
So the digit at the one’s place of  equation is =9

40. The sides of a right triangle whose hypotenuse is 17cm are _________ and _________.
Ans: 8 and 15
The  Pythagorean triplet sides are given by  equation .
If we take equation
equation
Then the value of equation will not be an integer.
So, we try to take 2equation=17.
Again, it  will not give an integer value for equation.
So, let us take equation,
Then equation
This implies equation=4.
Therefore, the required triplet is 2equation= equation, equation  and 17 .

41. equation
Ans:  1.4
equation

42.  equation
Ans.
1.728
equation

43. The cube of an odd number is always an _________ number.
Ans: always an odd number

44. The cube root of a number equation  is denoted by _________.
Ans. equation

45.  The least number by which 125 be multiplied to make it a perfect square is ……..
Ans. 5
Prime factorization  of 125=5×5×5
On grouping these factors in pair of equal factors, 125 = equation , we have only one 5 without a pair. So, 125 to be multiplied by 5 to make it a perfect square.

46. The least number by which 72 be multiplied to make it a perfect cube, is________
Ans. 3
Prime factorization of  72 is
72=2 x 2 x 2 x 3 x 3
Grouping the factors in triplets of equal factors, we get
equation
It is clear that 2 occurs as a prime factor of 72 thrice, but 3 occurs as a prime factor only twice. Thus, if we multiply 72 by 3, 3 will also occurs as a prime factor thrice and the product will be equation, which is a perfect cube.
Hence, the least number, which should be multiplied with 72 to get perfect cube, is 3.

47. The least number by which 72 be divided to make it a perfect cube, is________
Ans.  9
Prime factorization of  72 is
72=2 x 2 x 2 x 3 x 3
Grouping the factors in triplets of equal factors, we get
equation
Clearly, if we divide 72 by 3 x 3, the quotient would be 2 x 2 x 2, which is a perfect cube. Hence, the least number by which 72 be divided to make it, a perfect cube, is 9.

48. Cube of a number ending in 7 will end in the digit________
Ans.  3
The cubes of the numbers ending in digit 7 ends in digit 3.
For example: 7 x 7 x 7 = 343 , 17 x 17 x17 = 4913 etc.

True/False
In questions 49 to 86, state whether the statements are True or False.
 49.  The square of 86 will have 6 at the unit’s place.
Ans. True
The unit’s digit of the square of a number having digit at unit’s place as 4 or 6 is 6.

 50. The sum of two perfect squares is a perfect square.
Ans. False
e.g. 16 and 25 are the perfect squares, but 16 + 25 = 41 is not a perfect square.

51. The product of two perfect squares is a perfect square.
Ans. True

52.  There is no square number between 50 and 60.
Ans. True
Numbers between 50 and 60 are 51,52, 53, 54, 55, 56, 57, 58 and 59.
It is clear that  there is no square number between 50 and 60.

53. The square root of 1521 is 31.
Ans. False
Because equation .

54. Each prime factor appears 3 times in its cube.
Ans. True

55. The square of 2.8 is 78.4.
Ans. False
Since equation

56. The cube of 0.4 is 0.064.
Ans. True
As equation

57. The square root of 0.9 is 0.3.
Ans. False
As, the square of 0.3 = (0.3)2 = 0.3 x 0.3 =0.09

58. The square of every natural number is always greater than the number itself.
Ans. False
1 is a natural number and square of 1 is  1 which is equal to itself.

59. The cube root of 8000 is 200.
Ans. False 
Cube root of 8000 is 20.

60. There are five perfect cubes between 1 and 100.
Ans. False
There are eight perfect cubes between 1 and 100, i.e. 8, 27, 64, 125, 216, 343, 512 and 729.

61. There are 200 natural numbers between equation and equation .
Ans. 
True
equation
equation
Natural numbers s between equation and  equation.
Natural number between 10000 and 10201= 10201-10000-1=200.

62. The sum of first n odd natural numbers is equation .
Ans. True

63.  1000 is a perfect square.
Ans. False
1000 = 2 x 2 x 2 x 5 x 5 x 5 = 2 2 x 52x 2 x 5
Since there are two unpaired factors 2 and 5 , so  it is not a perfect square.

64. A perfect square can have 8 as its unit’s digit.
Ans. False

65. For every natural number equation , ( equationequation , equation) is a Pythagorean triplet.
Ans.
False
The  Pythagorean triplet sides are given by  equation .

66. All numbers of a Pythagorean triplet are odd.
Ans. False
Since equation . Hence 3,4, and 5 are Pythagorean triplet but 4 is an even number.

67. For an integer equation , equation is always greater than equation .
Ans. False
-2 is an integer  and equation which is less than equation .

68. If equation and equation are integers such that equation , then  equation .
Ans. False
Suppose , -1 and -2 are integers , then equation but  equation which is less than  equation.

69. Let  equation  and  equation be natural numbers. If  equation divides  equation, then equation divides equation.
Ans. True 
Let equation and equation be natural number such that  equation divides  equation , then equation .
equation
equation
equation divides  equation.

70.  If equation ends in 5 , then equation ends in 25.
Ans.
False
equation  end in 5  and equation= 42875 does not end in 25 .

71.  If equation ends in 9, then equation ends in 7.
Ans. 
equation ends in 9  and equation does not end in 7.

 

 

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