There are infinitely many rational numbers between any two integers or any two rational numbers. You can always find another one by dividing the interval further.
Always look at the denominator. If the denominator is 8, divide the space between each integer into 8 equal parts.
All fractions are rational numbers, but rational numbers also include negative values and integers (like -5, which can be written as -5/1), whereas we usually think of fractions as positive parts of a whole.
The process is exactly the same, but you move to the left of zero on the number line.
Mastering the visualization of numbers is simple with our NCERT Solutions for Class 8 Maths Ch 1: A Square and a Cube – Exercise 1.2. (Note: In standard NCERT, Chapter 1 is "Rational Numbers"). This exercise focuses on two critical skills: representing rational numbers on a number line and finding rational numbers between two given rational numbers.
Through the Class 8 NCERT Solutions for these chapters, learn to Plot and Compare Rational Numbers with Ease! The solutions to Exercise 1.2 are presented in a step-by-step approach to help students understand how to divide the space between integers into equal parts based on the denominator. These solutions follow CBSE Guidelines to help you improve your precision in geometry and number theory.
1.0Download NCERT Class 8 Maths Chapter 1 Ex 1.2 Solutions : Free PDF
You can download our simple NCERT Solutions for Class 8 Maths Chapter 1 Exercise 1.2 as a PDF. These solutions are prepared by ALLEN experts and are ideal for revision during the exam preparation.
Class 8 Maths Chapter 1 Ex 1.2 : Representing Rational Numbers
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2.0Detailed NCERT Class 8 Maths Chapter 1 Solutions of Exercise 1.2
1.Find the cube roots of 27000 and 10648.
Sol. Here,
2
27000
2
13500
2
6750
3
3375
3
1125
3
375
5
125
5
25
5
5
1
27000=2×2×2×3×3×3×5×5×5327000=2×3×5=30
2
10648
2
5324
2
2662
11
1331
11
121
11
11
1
10648=2×2×2×11×11×11310648=2×11=22
2. What number will you multiply by 1323 to make it a cube number?
Sol. Here,
3
1323
3
441
3
147
7
49
7
7
1
1323=3×3×3×7×7
To complete the triplet, one more 7 is required.
So, 1323 will be multiplied by 7 to make it a cube number.
So, the cube number =1323×7=9261
Hence, required number =73. State true or false. Explain your reasoning.
(i) The cube of any odd number is even.
(ii) There is no perfect cube that ends with 8.
(iii) The cube of a 2 -digit number may be a 3 -digit number.
(iv) The cube of a 2 -digit number may have seven or more digits.
(v) Cube numbers have an odd number of factors.
Sol. (i) The cube of any odd number is even.
(False)
Reason: The cube of an odd number is always odd, as
33=2753=12573=343
(ii) There is no perfect cube that ends with 8. (False)
Reason: The cubes of all the numbers ending with 2 at the unit place end with 8.
23=8123=1728223=10648
(iii) The cube of a 2 -digit number may be a 3 -digit number. (False)
Reason: Cube of a 2 -digit number may have a minimum of 4 digits to a maximum of 6 digits.
10 is the smallest 2-digit number, and 103=1000, which has 4 digits.
(iv) The cube of a 2 -digit number may have seven or more digits. (False)
Reason: Cube of a 2 -digit number may have at most 6 digits.
99 is the largest 2-digit number, and 993=970299, which is a 6 -digit number.
(v) Cube numbers have an odd number of factors. (False)
Reason: Cube numbers may have an odd as well as an even number of factors.
As 27=3×3×3 (odd no. of factors) 64=2×2×2×2×2×2 (even no. of factors)
4. You are told that 1331 is a perfect cube. Can you guess without factorisation what its cube root is? Similarly, guess the cube roots of 4913,12167 , and 32768.
Sol. To find the cube root of 1331,
3.01331
We divide the given number 1331 into two groups, starting from the right side, taking three digits in group 1.
331→ group 1
1→ group 2
331→ unit digit is 1
Hence, the cube roots of one's digit is 1
Group 2, i.e., 1 only, which is 13.
So, the cube roots of one's digit is 1 .
∵31331=11
4913
Group 1: 913
Group 2: 4
Unit digit of 913 is 3 .
We know that 3 comes at the unit's place when its cube root ends in 7 , as 7×7×7 = 343
So the unit digit of the cube root of 4913 =7
Group 2: 4
4 lies between 1 (i.e., 13 ) and 23 (i.e., 8 )
13<4<23
Taking the lower limit, the tens digit of the cube root of 4913 is 1 .
34913=17
(from (1) & (2))
12167
Group 1: 167
Unit digit = 7
So, unit digit of cube root of 12167=3 as 3×3×3=27
Group 2-12
8<12<2723<12<33
Taking the lower limit, the ten's digit of cube root =2
So, 312167=23
32768
Group 1: 768
Unit digit = 8
So, unit digit of cube root of 32768=2
as 2×2×2=8⇒38=2
From Group 2: 32
27<32<6433<32<43
Taking lower limit, ten's digit of the cube root of 32768 is 3 .
∴332768=32 (from (1) & (2))
5. Which of the following is the greatest?
Explain your reasoning.
(i) 673−663
(ii) 433−423
(iii) 672−662
(iv) 432−422
Sol. (i) 673−663=1+67×66×3
(ii) 433−423=1+43×42×3
(iii) 672−662=67+66=133
(iv) 432−422=43+42=85
From above we can see that 673−663 is the greatest as
(n+1)3−n3=1+(n+1)×3n(n+1)2−n2=n+n+1=2n+1
Very short answer type questions
1.Express 64 as the sum of eight odd numbers.
2. Find the number of digits in the square roots of the following
(i) 256
(ii) 4489
3. Without adding find the sum
1+3+5+7+9+11+13+15+17+19 SC13-0016
Find the square root of each of the following numbers by using the method of prime factorisation.
(i) 5184
(ii) 40000
(iii) 1444
5. Find the smallest number by which 252 must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.
6. By what smallest number must 180 be multiplied so that it becomes a perfect square? Also, find the square root of the number obtained.
7. Find the value of
(i) 40580
(ii) 72×338
8. Find the square root of the following fractions
(i) 2312126
(ii) 10225151
(iii) 361324
9. Evaluate : 7056
10. Find the cubes of
(i) 13
(ii) 1.3
(iii) 0.4
11. Which of the following numbers are perfect cubes?
(i) 4096
(ii) 392
12. Find the cubes of
(i) 0.6
(ii) - 3.1
13. Find the cubes of the following numbers.
(i) 133
(ii) 13−6
14. Which of the numbers are perfect cubes?
(i) 540
(ii) 900
15. Evaluate :
(i) (74)3
(ii) (1110)3
16. Find the cube roots of the following by prime factorization.
(i) 5832
(ii) 1728
17. Find the cube root of the
(i) -27000
(ii) -0.000001
SC14-0023
Short answer type questions
18. Find the value of 15625 and use it to find the value of 156.25+1.5625.
19. The area of a square field is 1014001m2. Find the length of one side of the field.
20. A general wishing to draw up his 64019 men in the form of a square found that he had 10 men extra. Find the number of men in the front row.
21. An army general arranges his soldiers in such a way that the number of rows is the same as the number of columns. In doing so, he finds that 55 soldiers are left out. If the total number of soldiers is 6455 , find the number of soldiers in each row.
22. Find the cost of erecting a fence around a square field whose area is 9 hectares, if fencing cost Rs. 3.50 per metre ( 1 hectare =10000m2 ).
23. What number when multiplied by itself will become 83.7225 ?
24. Find the smallest number by which 2560 must be multiplied so that the product is a perfect cube.
25. Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product.
26. By what smallest number 3645 be multiplied so that the product becomes a perfect cube?
27. By what smallest number 29160 be divided so that the quotient becomes a perfect cube?
Long answer type questions
28.By what smallest number 5184 be
(i) Multiplied
(ii) Divided,
so that the resulting number becomes a perfect cube.
29. By what smallest number should we divide 9000 so that the quotient becomes a perfect cube. Find the cube root of the quotient.
30. By what smallest number should we multiply 8788 so that the product becomes a perfect cube. Find the cube root of the product.
31. Find the smallest number by which 1323 must be multiplied so that the product is a perfect cube.
32. What is the smallest number by which 1600 must be divided so that the quotient is a perfect cube?
33. Find the smallest number by which 8788 must be divided so that the quotient is a perfect cube.
34. Find the cube roots of the following integers :
(i) -2744000
(ii) -474552
35. Evaluate : 38×125
36. Find the cube roots of the rational numbers :
(i) 33754913
(ii) 343−512
37. Find the value of the cube roots.
(i) 30.008
(ii) 31331−64
38. Evaluate :
38×17×17×17
39. Find the cube roots of
(i) 729×216
(ii) 359372744
(iii) 42197473
40. Find the cube root of 1372864.
41. Evaluate:
(i) 34096
(ii) 38000
42. Show that
(i) 3125×64=3125×364
(ii) 3216×(−343)=3216×3−343
43. Solve : (i) 32197216
(ii) 3512125
44. Find the cube root of 4.096 .
45. Is 12527 a cube of a fraction?
ANSWER KEY
Very short answer type questions
82=(1+3+5+7+9+11+13+15)
(i) 2
(ii) 2
100
(i) 72
(ii) 200
(iii) 38
7,42
5,30
(i) 94
(ii) 156
(i) 4119
(ii) 3154
(iii) 1918
9. 84
10.
(i) 2197
(ii) 2.197
11.
(i) Yes
(ii) No
(iii) 0.064
12.
(i) 0.216
(ii) –29.791
13.
(i) 2352637
(ii) 2197−216
14.
(i) No
(ii) No
15.
(i) 34364
(ii) 13311000
16.
(i) 18
(ii) 12
17.
(i) -30
(ii) -0.01
Short answer type questions
18. 13.75
19. 10201 metre
20. 253
21.80
22. Rs. 4200
23. 9.15
24. 25
25. 60, 60
26. 25
27.5
Long answer type questions
28.
(i) 9 (ii) 3
29. 9, 10
30. 2,26
31.7
32. 25
33. 4
34. (i) -140
(ii) -78
35. 10
36. (i) 1517
(ii) 7−8
37.
(i) 0.2
(ii) 11−4
38. 34
39. (i) 54
(ii) 3314
(iii) 1138
40. 76
41. (i) 16
(ii) 20
43.
(i) 136
(ii) 85
44. 1.6
45. Yes, 53
4.0Key Concepts of Chapter 1 Exercise 1.2
Representation on Number Line: To represent a rational number a/b, we divide the distance between two consecutive integers into b equal parts.
Rational Numbers between two Rational Numbers: There are countless (infinite) rational numbers between any two given rational numbers.
The Mean Method: One way to find a rational number between a and b is to find their average: (a+b)/2.
The Common Denominator Method: A faster way to find multiple rational numbers by making the denominators the same and then expanding the numerators.
5.0NCERT Solutions for Class 8 Maths Chapter 1 A Square and a Cube : All Exercises
6.0Key Features and Benefits of NCERT Solutions for Class 8 Maths Chapter 1 Exercise 1.2
Clear Conceptual Explanation of Number Placement: The solutions explain how numbers exist between any two given values, helping learners understand the infinite nature of the number line in a simple and structured way.
Step-by-Step Logical Methodology: Each solution follows a systematic approach, guiding learners through efficient techniques like using common denominators instead of lengthy calculations.
Improved Accuracy in Representation: Emphasis is placed on precise placement of numbers on the number line, ensuring mathematical correctness and clarity in answers.
Time-Saving Problem-Solving Techniques: The methods used help reduce unnecessary steps, allowing learners to solve questions faster and more efficiently.
Error Reduction Through Structured Solutions: Clear and organised explanations minimise confusion and help avoid common mistakes while working with fractions and intervals.
Strengthens Analytical Thinking: Encourages learners to think logically about number relationships and apply the most effective strategy to solve problems.
Builds a Strong Mathematical Foundation: Develops essential skills that are crucial for advanced topics like rational numbers, algebra, and coordinate geometry.