Exercise 1.1

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NCERT Solutions Class 8 Maths Chapter 1 A Square and a Cube - Exercise 1.1

Class 8 Mathematics Chapter 1 - A Square and A Cube - Exercise 1.1. The goal of the exercise is to learn how to determine whether a number is a square number, understand the ways in which numbers behave, and apply prime factorisation to find out if a number is a perfect square. The first chapter of Mathematics will enable students to identify patterns in numbers, including patterns involving numbers ending in certain digits, relationships among odd numbers, and their connections to the sums of squares.

Step-by-step NCERT Solutions for Class 8 Exercise 1.1 equip students with a process for establishing a strong logical framework that supports accurate calculation and clear understanding of concepts. The step-by-step solutions are based on the current CBSE syllabus which means they can be used as references for homework help and exam preparation. These solutions can also help you determine what the smallest number you can multiply to produce a perfect square is or find square patterns. The detailed explanations will make complex arithmetic simple. Use these solutions to enhance your confidence and achieve high grades in your mathematics assessments.

1.0Download Class 8 Maths Chapter 1 Ex 1.1 NCERT Solutions PDF

Access our easy-to-understand NCERT Solutions for Class 8 Maths Chapter 1 Exercise 1.1 as a downloadable PDF. This allow you to study offline and reference the material whenever needed.

Class 8 Maths Chapter 1 A Square and a Cube : Exercise 1.1

Download PDF

2.0Detailed NCERT Class 8 Maths Chapter 1 Solutions of Exercise 1.1

1.Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

Sol. (i) 2032 is not a perfect square, as a number ending with 2 cannot be a perfect square. (ii) 2048 is not a perfect square, as a number ending with 8 cannot be a perfect square. (iii) 1027 is not a perfect square, as a number ending with 7 cannot be a perfect square. (iv) 1089 ends in 9 at the unit's place. Hence, it is a perfect square. 2. Which one among $6 4^{2}, 10 8^{2}, 29 2^{2}, 3 6^{2}$ has the last digit 4 ?

Sol. (i) Unit's digit of 64 is 4 $∴ 4^{2} = 4 \times 4 = 16$ (last digit $= 6$ ) (ii) Unit's digit of 108 is 8 $∴ 8^{2} = 8 \times 8 = 64$ (last digit $= 4$ ) (iii) Unit's digit of 292 is 2 $∴ 2^{2} = 2 \times 2 = 4$ (iv) Unit's digit of $36 = 6$ $∴ 6^{2} = 6 \times 6 = 36$ (last digit $= 6$ ) Hence, the numbers whose squares end in 4 are $10 8^{2}$ and $29 2^{2}$ . 3. Given $12 5^{2} = 15625$ , what is the value of 126²? (i) $15625 + 126$ (ii) $15625 + 2 6^{2}$ (iii) $15625 + 253$ (iv) $15625 + 251$ (v) $15625 + 5 1^{2}$

Sol. Here, $12 6^{2} = (125 + 1)^{2}$ $= (125)^{2} + 2 \times 125 \times 1 + (1)^{2}$ [Using identity $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ ] $= 15625 + 250 + 1$ $= 15625 + 251$ So, the value of $12 6^{2}$ is (iv) option, i.e., $15625 + 251$ . 4. Find the length of the side of a square whose area is $441 m^{2}$ .

Sol. Area of square $=$ side × side $= 441$ $\Rightarrow$ side $^{2} = 441$ $\Rightarrow$ side $= 441$

3	441
3	147
7	49
7	7
	1

$441 = \underline{3 \times 3} \times \underline{7 \times 7} 441 = 3 \times 7 = 21 Side of square = 21 m$

Find the smallest square number that is divisible by each of the following numbers: 4,9 , and 10 LCM $= 2 \times 2 \times 3 \times 3 \times 5 = 180$ Prime factorisation of $180 = 2 \times 2 \times 3 \times 3 \times 5$ 5 is not in pairs, so 180 is not a square number.
Sol. To find the required smallest square number, we will find the least number divisible by each of 4,9 , and 10 , i.e., LCM of 4,9 , and 10 .

2	4,	9,	10
2	2,	9,	5
3	1,	9,	5
3	1,	3,	5
5	1,	1,	5
	1,	1,	1

In order to get a perfect square, we will multiply 180 by 5 . So, the required smallest square number is 900. 6. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

Sol.

2	9408
2	4704
2	2352
2	1176
3	588
3	294
3	147
7	49
7	7
	1

$9408 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 7 \times 7$ All prime factors of 9408 are arranged in pairs except 3. So, we multiply 9408 by 3 to make it a perfect square.

Perfect square $= 9408 \times 3 = 28224$ Now, $28224 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7$ $= 2 \times 2 \times 3 \times 3 \times 7$ $= 252$ 7. How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100

Sol. (i) Numbers lying between $1 6^{2}$ and $1 7^{2}$

$= 2 \times 16 = 32$

(ii) Numbers lying between $9 9^{2}$ and $10 0^{2}$

8.In the following pattern, fill in the missing numbers: $1^{2} + 2^{2} + 2^{2} = 3^{2}$ $2^{2} + 3^{2} + 6^{2} = 7^{2}$ $3^{2} + 4^{2} + 1 2^{2} = 1 3^{2}$ $4^{2} + 5^{2} + 2 0^{2} = (_)^{2}$ $9^{2} + 1 0^{2} + (_)^{2} = (_)^{2}$

Sol. $1^{2} + 2^{2} + 2^{2} = 3^{2}$ $2^{2} + 3^{2} + 6^{2} = 7^{2}$ $3^{2} + 4^{2} + 1 2^{2} = 1 3^{2}$ $4^{2} + 5^{2} + 2 0^{2} = 2 1^{2}$ $9^{2} + 1 0^{2} + 9 0^{2} = 9 1^{2}$

9.How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

Sol. Big squares in a row $= 9$ Big squares in a column $= 9$ Tiny squares in a big square $= 25$ ∴ Total tiny squares $= 9 \times 9 \times 25 = 2025$ Now prime factorisation of 2025 $= 3 \times 3 \times 3 \times 3 \times 5 \times 5$

Multiple choice questions

1.Which of the following numbers is a perfect square? (1) 141 (2) 196 (3) 124 (4) 222

2. A perfect square number can never have the digit ..... at the unit's place. (1) 1 (2) 4 (3) 8 (4) 9

3. What least number must be multiplied to 12288 so that the product becomes a perfect square? (1) 2 (2) 3 (3) 4 (4) 5

4. The smallest number by which 12348 must be divided to obtain a perfect square is (1) 3 (2) 4 (3) 5 (4) 7

5. Sum of the first $n$ odd natural numbers is (1) $2 n + 1$ (2) $n^{2}$ (3) $n^{2} - 1$ (4) $2 n^{2} + 1$

6. The value of the expression $1 + 23 1 + 24 1 + 25 26 \times 28 + 1$ is equal to (1) 24 (2) 25 (3) 26 (4) None of these

7. The value of $214 + 130 - 88 - 44 + 25$ (1) 14 (2) 15 (3) 16 (4) 17

8. $4 \frac{57}{196} =$ ? (1) $2 \frac{1}{14}$ (2) $2 \frac{3}{14}$ (3) $2 \frac{5}{14}$ (4) $2 \frac{9}{14}$

9. $0.09 =$ ? (1) 0.3 (2) 0.03 (3) 0.33 (4) None of these 10. $0.9 \times 1.6 =$ ? (1) 0.12 (2) 1.2 (3) 0.75 (4) 12

11. If $\frac{16}{49} = \frac{n}{49}$ then $n =$ (1) 4 (2) 7 (3) 16 (4) 28

12. $5625 =$ ? (1) 55 (2) 65 (3) 75 (4) 85

13. $1 + 3 + 5 + 7 + \dots\dots\dots + 49$ should be equal to (1) $4 9^{2}$ (2) $2 4^{2}$ (3) $2 5^{2}$ (4) None of these 14. Write $(\frac{27}{125})$ in index form (1) $(\frac{3}{5})^{3}$ (2) $(\frac{5}{3})^{3}$ (3) $(\frac{4}{5})^{3}$ (4) None of these

15. Which one of the following numbers is not a perfect cube? 64, 216, 343, 256 (1) 64 (2) 216 (3) 343 (4) 256

16. Which of the following numbers is a perfect cube? (1) 1525 (2) 1728 (3) 1458 (4) 3993

SC13-001317. Which of the following numbers is not a perfect cube? (1) 2197 (2) 512 (3) 2916 (4) 343

18. The smallest natural number by which 25 must be multiplied to get a perfect cube is (1) 5 (2) 20 (3) 25 (4) None of these

19. The smallest natural number by which 1296 be divided to get a perfect cube is (1) 16 (2) 6 (3) 60 (4) None of these

20. Which of the following numbers are the cube of a negative whole number? -64, -2197, -1056, -3888 (1) $- 64, - 2197$ (2) $- 1056, - 3888$ (3) $- 64, - 1056$ (4) -2197, -3888

21. The value of $(3.1)^{3}$ is (1) 27.971 (2) 29.791 (3) 29.971 (4) 27.197

22. What is the least number by which 8640 is divided, to get the quotient as a perfect cube number? (1) 6 (2) 7 (3) 5 (4) 8

23. The value of $(- 474552)^{1/3}$ is (1) -68 (2) -58 (3) -78 (4) 88

24. The value of $3 - a^{3} \times 3 - b^{3}$ is (1) a (2) b (3) ab (4) None of these

25. $3 \frac{- 16}{128}$ is equal to (1) $\frac{4}{12}$ (2) $\frac{1}{2}$ (3) $- \frac{1}{2}$ (4) None of these

26. The value of $3 5 \times 25$ is (1) 5 (2) 25 (3) 125 (4) None of these

27. Which of the following numbers are cubes of fractions? $\frac{27}{64}, \frac{125}{128}, 0.001331, 0.04$ (1) $\frac{27}{64}, 0.001331$ (2) $\frac{125}{128}$ (3) 0.04 (4) None of these

28. The value of $[64 \times (- 2744)]^{1/3}$ is (1) 56 (2) -56 (3) 65 (4) -65

29. Statement-1: Each Side length of cube of volume 512 cu . metres is 8 m . Statement-2: Least number to be multiplied with 864 to make it a perfect cube is 2 . (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both the statements are true. (4) Both the statements are false.

30. Statement-1: $35832 = 18$

Statement-2: $3 5.832 = 0.18$ (1) Statement-1 is true and statement-2 is false. (2) Statement-1 is false and statement-2 is true. (3) Both the statements are true. (4) Both the statements are false.

True or false

31. The number of digits in a perfect square is even.

32. The square of a prime number is prime.

33. The number 22222 is a perfect square.

34. The difference of two perfect squares is a perfect square.

35. The product of two perfect squares is a perfect square.

36. Cube of all even natural numbers are even.

37. Cube of all odd natural numbers are odd.

38. Cubes of negative integers are negative.

39. 8640 is not a perfect cube.

40. There is no perfect cube which ends in 4.

Fill in the blanks

41. The square of an even number is $____$ 42. The square of an odd number is $____$ 43. The square of a proper fraction is $____$ than the given fraction.

44. $n^{2} =$ the sum of first $n$ $____$ natural numbers.

45. The value of $6.25$ is $____$ 46. The sum of any two consecutive $____$ numbers is a perfect square number.

47. $4 8^{2} = (\dots\dots .. \times 12)^{2}$

48. The value of $\frac{1.21 \times 0.9}{1.1 \times 0.11}$ is $____$ 49. $(\frac{6}{13})^{2} = \frac{6 ^{2}}{□}$ □ 50. $(92)^{2} - (91)^{2} =$ $____$ 51. A number raised to power 3 is called $____$ .

52. Out of the following 8, 81, 25, 100 the perfect cube is $____$ .

53. Out of the following 8, 64, 216, 343 the cube of odd natural number is $____$ . 54. The cubes of all even number between 1 and 5 are $____$ .

55. Out of the following 27, 125, 343, 512, 729 the cube of even natural number is $____$ .

56. The numbers whose cube and cube root both are equal are $____$ .

57. The smallest natural number by which 36 must be multiplied to get a perfect cube is $____$ .

58. The smallest natural number by which 2401 must be divided to get a perfect cube is $____$ .

59. The value of $3 \frac{- 27}{125}$ is $____$ . 60. The value of $3 - 512 \times 38$ is $____$ .

61. The cube root of - 1 is $____$ . 62. $3 \frac{0.64}{10} =$ $____$ .

63. If $\frac{3 0.512}{x} = 31000000$ , then the value of $x$ is $____$ .

64. The cube of 1.1 is $____$ .

Puzzle 65. Using only square and cube numbers less than or equal to 100, can you fill in the circles to make these sums true? You can only use each number once and you must use all the numbers. (i)

(ii)

(iii)

ANSWER KEY

Multiple choice questions

Question	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Answer	$2$	$3$	$2$	$4$	$2$	$1$	$2$	$1$	$1$	$2$	$4$	$3$	$3$	$1$	$4$
Question	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Answer	$2$	$3$	$1$	$2$	$1$	$2$	$3$	$3$	$3$	$3$	$1$	$1$	$2$	$3$	$1$

True or false 31. False

32. False

33. False

34. False

35. True

36. True

37. True

38. True

39. True

40. False Fill in the blanks 41. even

42. odd

43. smaller

44. odd

45. 2.5

46.triangular

47.4

48.3

$49.1 3^{2}$

50. 183

51. Cube of that number

52.8

53.343

54. 8, 64

55.512

56. $- 1, 1$

57.6

58.7

59. –3/5

60. -16

61. -1

62. 0.4

63. 0.008

64. 1.331

Puzzle 65. (i) $9 + 16 = 25$ (ii) $36 + 49 = 4 + 81$ (iii) $1 + 8 + 27 + 64 = 100$

3.0Key Concepts of Chapter 1 A Square and a Cube Exercise 1.1

Characteristics of Square Numbers: Recognize that square numbers will always have a 0, 1, 4, 5, 6, or 9 digit at the end.
Prime Factorization: Take a number, break it down to its prime factors and check for pairs.
Finding the Lowest Common Multiple of Perfect Squares: Find the smallest number by which you can multiply or divide a number to create a perfect square.
Patterns Among Squares: The squares of both odd and even numbers will show the same number of zeros at the end of their squares.

4.0NCERT Solutions for Class 8 Maths Chapter 1 A Square and a Cube : All Exercises

Chapter 1 A Square and a Cube - Exercise 1.1

Chapter 1 A Square and a Cube - Exercise 1.2

5.0Key Features and Benefits of NCERT Solutions Class 8 Maths Chapter 1 Exercise 1.1

Structured Approach to Prime Factorisation: The solutions follow a clear, step-by-step method to break down numbers into their prime factors, helping learners understand the logic behind factorisation rather than memorising steps.
Enhanced Understanding of Square Number Patterns: By analysing number patterns, learners develop the ability to recognise properties of perfect squares, improving their conceptual clarity and pattern recognition skills.
Aligned with CBSE Guidelines: The content strictly follows the latest CBSE curriculum, using appropriate methods and terminology to ensure relevance and accuracy for Class 8 students.
Faster Identification of Perfect and Non-Perfect Squares: The techniques taught enable quick identification of square numbers, saving time during problem-solving and improving efficiency in exams.
Reduction of Calculation Errors: Stepwise explanations minimise confusion and help learners avoid common mistakes while solving factorisation and square-related problems.
Builds a Strong Foundation for Higher Concepts: Mastery of these basics supports future topics like square roots, algebra, and number systems, ensuring long-term mathematical understanding.

NCERT Class 8 Maths Ch. 1 A Square and A Cube Other Exercises:-

Exercise 1.1

Exercise 1.2

NCERT Solutions Class 8 Maths All Chapters:-

Chapter 1 - A Square and a Cube

Chapter 2 - Power play

Chapter 3 - A story of Numbers

Chapter 4 - Quadrilaterals

Chapter 5 - Number Play

Chapter 6 - We distibute, yet thinmgs multiply

Chapter 7 - Proportional Reasoning

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NCERT Solutions Class 8 Maths Chapter 1 A Square and a Cube - Exercise 1.1

Class 8 Mathematics Chapter 1 - A Square and A Cube - Exercise 1.1. The goal of the exercise is to learn how to determine whether a number is a square number, understand the ways in which numbers behave, and apply prime factorisation to find out if a number is a perfect square. The first chapter of Mathematics will enable students to identify patterns in numbers, including patterns involving numbers ending in certain digits, relationships among odd numbers, and their connections to the sums of squares.

Step-by-step NCERT Solutions for Class 8 Exercise 1.1 equip students with a process for establishing a strong logical framework that supports accurate calculation and clear understanding of concepts. The step-by-step solutions are based on the current CBSE syllabus which means they can be used as references for homework help and exam preparation. These solutions can also help you determine what the smallest number you can multiply to produce a perfect square is or find square patterns. The detailed explanations will make complex arithmetic simple. Use these solutions to enhance your confidence and achieve high grades in your mathematics assessments.

1.0Download Class 8 Maths Chapter 1 Ex 1.1 NCERT Solutions PDF

Access our easy-to-understand NCERT Solutions for Class 8 Maths Chapter 1 Exercise 1.1 as a downloadable PDF. This allow you to study offline and reference the material whenever needed.

Class 8 Maths Chapter 1 A Square and a Cube : Exercise 1.1

Download PDF

2.0Detailed NCERT Class 8 Maths Chapter 1 Solutions of Exercise 1.1

1.Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

Sol. (i) 2032 is not a perfect square, as a number ending with 2 cannot be a perfect square. (ii) 2048 is not a perfect square, as a number ending with 8 cannot be a perfect square. (iii) 1027 is not a perfect square, as a number ending with 7 cannot be a perfect square. (iv) 1089 ends in 9 at the unit's place. Hence, it is a perfect square. 2. Which one among $6 4^{2}, 10 8^{2}, 29 2^{2}, 3 6^{2}$ has the last digit 4 ?

Sol. (i) Unit's digit of 64 is 4 $∴ 4^{2} = 4 \times 4 = 16$ (last digit $= 6$ ) (ii) Unit's digit of 108 is 8 $∴ 8^{2} = 8 \times 8 = 64$ (last digit $= 4$ ) (iii) Unit's digit of 292 is 2 $∴ 2^{2} = 2 \times 2 = 4$ (iv) Unit's digit of $36 = 6$ $∴ 6^{2} = 6 \times 6 = 36$ (last digit $= 6$ ) Hence, the numbers whose squares end in 4 are $10 8^{2}$ and $29 2^{2}$ . 3. Given $12 5^{2} = 15625$ , what is the value of 126²? (i) $15625 + 126$ (ii) $15625 + 2 6^{2}$ (iii) $15625 + 253$ (iv) $15625 + 251$ (v) $15625 + 5 1^{2}$

Sol. Here, $12 6^{2} = (125 + 1)^{2}$ $= (125)^{2} + 2 \times 125 \times 1 + (1)^{2}$ [Using identity $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ ] $= 15625 + 250 + 1$ $= 15625 + 251$ So, the value of $12 6^{2}$ is (iv) option, i.e., $15625 + 251$ . 4. Find the length of the side of a square whose area is $441 m^{2}$ .

Sol. Area of square $=$ side × side $= 441$ $\Rightarrow$ side $^{2} = 441$ $\Rightarrow$ side $= 441$

3	441
3	147
7	49
7	7
	1

$441 = \underline{3 \times 3} \times \underline{7 \times 7} 441 = 3 \times 7 = 21 Side of square = 21 m$

Find the smallest square number that is divisible by each of the following numbers: 4,9 , and 10 LCM $= 2 \times 2 \times 3 \times 3 \times 5 = 180$ Prime factorisation of $180 = 2 \times 2 \times 3 \times 3 \times 5$ 5 is not in pairs, so 180 is not a square number.
Sol. To find the required smallest square number, we will find the least number divisible by each of 4,9 , and 10 , i.e., LCM of 4,9 , and 10 .

2	4,	9,	10
2	2,	9,	5
3	1,	9,	5
3	1,	3,	5
5	1,	1,	5
	1,	1,	1

In order to get a perfect square, we will multiply 180 by 5 . So, the required smallest square number is 900. 6. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

Sol.

2	9408
2	4704
2	2352
2	1176
3	588
3	294
3	147
7	49
7	7
	1

$9408 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 7 \times 7$ All prime factors of 9408 are arranged in pairs except 3. So, we multiply 9408 by 3 to make it a perfect square.

Perfect square $= 9408 \times 3 = 28224$ Now, $28224 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7$ $= 2 \times 2 \times 3 \times 3 \times 7$ $= 252$ 7. How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100

Sol. (i) Numbers lying between $1 6^{2}$ and $1 7^{2}$

$= 2 \times 16 = 32$

(ii) Numbers lying between $9 9^{2}$ and $10 0^{2}$

8.In the following pattern, fill in the missing numbers: $1^{2} + 2^{2} + 2^{2} = 3^{2}$ $2^{2} + 3^{2} + 6^{2} = 7^{2}$ $3^{2} + 4^{2} + 1 2^{2} = 1 3^{2}$ $4^{2} + 5^{2} + 2 0^{2} = (_)^{2}$ $9^{2} + 1 0^{2} + (_)^{2} = (_)^{2}$

Sol. $1^{2} + 2^{2} + 2^{2} = 3^{2}$ $2^{2} + 3^{2} + 6^{2} = 7^{2}$ $3^{2} + 4^{2} + 1 2^{2} = 1 3^{2}$ $4^{2} + 5^{2} + 2 0^{2} = 2 1^{2}$ $9^{2} + 1 0^{2} + 9 0^{2} = 9 1^{2}$

9.How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

Sol. Big squares in a row $= 9$ Big squares in a column $= 9$ Tiny squares in a big square $= 25$ ∴ Total tiny squares $= 9 \times 9 \times 25 = 2025$ Now prime factorisation of 2025 $= 3 \times 3 \times 3 \times 3 \times 5 \times 5$