The chapter cubes and cube roots is an integral part of the mathematics syllabus of class 8. To understand the topic fully, students must take the help of precise and easy-to-understand NCERT solutions.
S. No. |
Topic |
7.1 |
Introduction |
7.2 |
Cubes |
7.2.1 |
Some interesting patterns |
7.2.2 |
Smallest multiple that is a perfect cube |
7.3 |
Cube roots |
7.3.1 |
Cube root through prime factorization method |
7.3.2 |
Cube root of a cube number |
Chapter 7, cubes and cube roots, helps students learn the methods of calculation of cubes and cube roots. It is easy, for students, to solve the NCERT questions on mathematical concepts, by making use of the solutions on the MSVGo app.
The different exercises in chapter 7 assist students to learn and understand the details of the concept of cubes and cube roots. Two different exercises focus on different problems.
There are two exercises in NCERT solutions for class 8 maths chapter 7, cubes and cube roots. Let us start with these exercises:
1. Which of the following numbers are not perfect cubes?
216
Solution:
Resolve 216 into its prime factors.
216 = 2*2*2*3*3*3
Make triplets of equal factors.
216 = (2*2*2)*(3*3*3)
= (2*3)
= 6
As 216 is the cube of 6, it is a perfect cube.
Solution:
Resolve 128 into its prime factors.
128 = 2*2*2*2*2*2*2
Make triplets of equal factors.
128 = (2*2*2)*(2*2*2)*2
As 128 cannot be grouped into triplets of equal factors, it is not a perfect cube.
1000
Solution:
Resolve 1000 into its prime factors.
1000 = 2*2*2*5*5*5
Make triplets of equal factors.
1000 = (2*2*2)*(5*5*5)
As 1000 is the cube of 10, it is a perfect cube.
100
Solution:
Resolve 100 into its prime factors.
100 = 2*2*5*5
As 100 cannot be grouped into triplets of equal factors, it is not a perfect cube.
46656
Solution:
Resolve 46656 into its prime factors.
1000 = 2*2*2*2*2*2*3*3*3*3*3*3
Make triplets of equal factors.
1000 = (2*2*2)*(2*2*2)*(3*3*3)*(3*3*3)
As 46656 is a cube of 36, it is a perfect cube.
2. Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.
243
Solution:
Resolve the given number into its prime factors.
243 = 3*3*3*3*3
Make triplets of equal factors.
243 = (3*3*3)*3*3
Hence, 243 should be multiplied by 3 to get a perfect cube.
256
Solution:
Resolve the given number into its prime factors.
256 = 2*2*2*2*2*2*2*2
Make triplets of equal factors.
256 = (2*2*2)*(2*2*2)*2*2
Hence, 256 should be multiplied by 2 to get a perfect cube.
72
Solution:
Resolve the given number into its prime factors.
72 = 2*2*2*3*3
Make triplets of equal factors.
72 = (2*2*2)* (3*3)
Hence, 72 should be multiplied by 3 to get a perfect cube.
675
Solution:
Resolve the given number into its prime factors.
675 = 3*3*3*5*5
Make triplets of equal factors.
675 = (3*3*3)* (5*5)
Hence, 675 should be multiplied by 5 to get a perfect cube.
100
Solution:
Resolve the number into its prime factors.
100 = 2*2*5*5
To make triplets of equal factors, 2 and 5 are needed.
Hence, 100 should be multiplied by 10 to get a perfect cube.
3. Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.
81
Solution:
Resolve the given number into its prime factors.
81 = 3*3*3*3
Make triplets of equal factors.
81 = (3*3*3)* 3
Hence, 81 should be divided by 3 to get a perfect cube.
128
Solution:
Resolve the given number into its prime factors.
128 = 2*2*2*2*2*2*2
Make triplets of equal factors.
128 = (2*2*2)* (2*2*2)*2
Hence, 128 should be divided by 2 to get a perfect cube.
135
Solution:
Resolve the given number into its prime factors.
135 = 3*3*3*5
Make triplets of equal factors.
135 = (3*3*3)* 5
Hence, 135 should be divided by 5 to get a perfect cube.
192
Solution:
Resolve the given number into its prime factors.
192= 2*2*2*2*2*2*3
Make triplets of equal factors.
192 = (2*2*2)* (2*2*2)*3
Hence, 192 should be divided by 3 to get a perfect cube.
704
Solution:
Resolve the given number into its prime factors.
704 = 2*2*2*2*2*2*11
Make triplets of equal factors.
704 = (2*2*2)* (2*2*2)*11
Hence, 704 should be divided by 11 to get a perfect cube.
4. Parikshit is making a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?
Solution:
Volume of the cuboid = 5*2*5 = 50
Resolve 50 into its prime factors.
50 = 2*5*5
2, 5, and 5 cannot be grouped into triplets of equal factors. We therefore have to multiply 50, by 2*2*5 = 20, to get a perfect cube.
Hence, Parikshit will require 20 such cuboids to make a cube.
1.Find the cube root of each of the following numbers by the prime factorization method.
64
Resolve the given number into its prime factors.
64 = 2*2*2*2*2*2
Make triplets of equal factors.
64 = (2*2*2)*(2*2*2)
2*2 = 4
Hence, 4 is the cube root of 64.
512
Resolve the given number into its prime factors.
512 = 2*2*2*2*2*2*2*2*2
Make triplets of equal factors.
512 = (2*2*2)*(2*2*2)*(2*2*2)
2*2*2 = 8
Hence, 8 is the cube root of 512.
10648
Resolve the given number into its prime factors.
10648 = 2*2*2*1 1*1 1*11
Make triplets of equal factors.
10648 = (2*2*2)*(11*11*11)
2*11 = 22
Hence, 22 is the cube root of 10648.
27000
Resolve the given number into its prime factors.
27000 = 2*2*2*3*3*3*3*5*5*5
Make triplets of equal factors.
27000 = (2*2*2)*(3*3*3)*(5*5*5)
(2*3*5) = 30
Hence, 30 is the cube root of 27000.
15625
Resolve the given number into its prime factors.
15625 = 5*5*5*5*5*5
Make triplets of equal factors.
15625 = (5*5*5 )*(5*5*5)
(5*5) = 25
Hence, 25 is a cube root of 15625.
13824
Resolve the given number into its prime factors.
13824 = 2*2*2*2*2*2*2*2*2*3*3*3
Make triplets of equal factors.
13824 = (2*2*2)*(2*2*2)*(2*2*2)*(3*3*3)
(2*2*2*3) = 24
Hence, 24 is the cube root of 13824.
110592
Resolve the given number into its prime factors.
110592 = 2*2*2*2*2*2*2*2*2*2*2*2*3*3*3
Make triplets of equal factors.
110592 = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)*(3*3*3)
(2*2*2*2*3) = 48
Hence, 48 is the cube root of 110592.
46656
Resolve the given number into its prime factors.
46656 = 2*2*2*2*2*2*3*3*3*3*3*3
Make triplets of equal factors.
46656 = (2*2*2)*(2*2*2)*(3*3*3)*(3*3*3)
(2*2*3*3) = 36
Hence, 36 is the cube root of 46656.
175616
Resolve the given number into its prime factors.
175616 = 2*2*2*2*2*2*2*2*2*7*7*7
Make triplets of equal factors.
175616 = (2*2*2)*(2*2*2)*(2*2*2)*(7*7*7)
(2*2*2*7) = 56
Hence, 56 is the cube root of 175616.
91125
Resolve the given number into its prime factors.
91125 = 3*3*3*3*3*3*3*5*5*5
Make triplets of equal factors.
91125 = (3*3*3)*(3*3*3)*(5*5*5)
(3*3*5) = 45
Hence, 45 is the cube root of 91125.
2. State true or false.
The cube of any odd number is even. False
A perfect cube does not end with two zeros. True
If a cube of a number ends with 5, then its cube ends with 25. False
There is no perfect cube that ends with 8. False
The cube of a two-digit number may be a three-digit number. False
The cube of a two-digit number may have seven or more digits. False
The cube of a single-digit number may be a single-digit number. True
3. You are told that 1331 is a perfect cube. Can you guess without factorization what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.
Solution:
1331
Group the digits of the number to get 1 and 331. Here, 1 is the unit digit of the cube root of 1331.
The cube of 1 matches with the number of the second group. Take the tens digit of the cube root as the unit place of the smallest number.
Only the number 1 has 1 in the unit place of its cube.
Hence, ∛1331 = 11.
4913
Group the digits of the number to get 4 and 913. Here, 7 is a unit digit of the cube root of 4913. We know, 1^{3} = 1, 2^{3} = 8, and 1 > 4 > 8.
Also, 1 is the tens digit of the cube root.
Hence, ∛4913 = 17.
12167
Group the digits to get 12 and 167. Here, the unit digit of the cube is 7, the unit digit of the cube root is 3
Now, 2^{3} = 8, and 3^{3 }= 27, 8 > 12 > 27.
Thus, 2 is the tens digit of the cube root.
Hence, ∛12167 = 23.
32768
Group the digits to get 32 and 768. Here, the unit digit of the cube is 8, the unit digit of the cube root is 2,
Also 3^{3} = 27, and 4^{3} = 64 , 27 > 32 > 64.
Thus, 3 is the tens digit of the cube root.
Hence, ∛32768 = 32.
For the ease of understanding, students require access to detailed and accurate solutions to all questions of the chapter cubes and cube roots of class 8 maths. The MSVGo app provides these solutions for the benefit of learners. It also provides the bonus of creative maths games and interschool competitions.
1. How many questions are there in chapter 7?
In chapter 7, there are two exercises carrying 4 and 3 questions, respectively.
2. What is meant by cubes and cube roots?
The cube is the product obtained when a number is multiplied by itself thrice. The cube root of any number is the root number or factor that when multiplied by itself 3 times gives us the original number.
3. Can I get NCERT solutions for class 8 maths chapter 7 online?
Yes, you can easily NCERT solutions for class 8 maths, chapter cubes and cube roots on the MSVGo app. It offers precise and detailed solutions for all NCERT exercise questions.
4. Can I download NCERT maths class 8 solutions for the chapter cubes and cube roots for free?
You can download NCERT maths class 8 solutions for the chapter cubes and cube roots for free using the MSVGo platform. You can also download the free PDF for easy and quick access while studying the chapter.