NCERT Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots

NCERT Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots

Edited By Komal Miglani | Updated on Apr 17, 2025 09:29 AM IST

In our childhood, many of us have tried to stack small cubes to create a bigger cube box. Also, some of you have already come across many perfect shapes-related puzzles like Rubik's Cube. These are all based on the idea of Cubes. In mathematics, when you multiply a number by itself three times, then the result is the cube of that number. For example cube of 2 is (2 × 2 × 2) = 8. Also, the cube root is the number which is used to make a cube, i.e. cube root of 8 is 2. In the 6th chapter of the NCERT Class 8 Maths, you will find Cubes and Cube Roots.

This Story also Contains
  1. Cubes and Cube Roots Class 8 Questions And Answers PDF Free Download
  2. Cubes and Cube Roots Class 8 Solutions - Important Formulae
  3. Cubes and Cube Roots Class 8 NCERT Solutions
  4. NCERT Solutions For Class 8 Maths - Chapter Wise
  5. Importance of Solving NCERT Questions of Class 8 Maths Chapter 6
  6. NCERT Class 8 Maths Solutions: Subject Wise
  7. NCERT Books and Syllabus for Class 8
NCERT Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots
NCERT Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots

This article on NCERT solutions for class 8 Maths Chapter 6 Cubes and Cube Roots offers clear and step-by-step solutions for the exercise problems in the NCERT Books for class 8 Maths. Students who are in need of Cubes and Cube Roots class 8 solutions will find this article very useful. It covers all the important Class 8 Maths Chapter 6 question answers. These Cubes and Cube Roots class 8 ncert solutions are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 8 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions.

Cubes and Cube Roots Class 8 Questions And Answers PDF Free Download

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Cubes and Cube Roots Class 8 Solutions - Important Formulae

Cube Root Formula: For any number $m$, which can be expressed as the product of any number three times, $m = n × n × n = n^3$

The cube root of $m$ is denoted as: $\sqrt[3]{m}=n$

Methods of Finding Cube Roots:

  • Prime Factorization Method

  • Estimation Method

Cubes and Cube Roots Class 8 NCERT Solutions

Class 8 maths chapter 6 question answer - Topic 6.2 Cubes

Q(i) Find the one’s digit of the cube of each of the following numbers.

3331

Answer: The detailed solution for the above-mentioned question is as follows,

Since the given number ends with 1, so the one’s digit of the cube of 3331 will be 1.

Q(ii) Find the one’s digit of the cube of each of the following numbers.

8888

Answer: The detailed solution for the above-mentioned question is as follows

Since the given number ends with 8, so the one’s digit of the cube of 8888 will be 2.

Q(iii) Find the one's digit of the cube of each of the following numbers.

149

Answer: The detailed solution for the above-mentioned question is as follows,

Since the given number has 9 at units place, so the one’s digit of the cube of 149 will be 9.

Q(iv) Find the one’s digit of the cube of each of the following numbers.

1005

Answer: The detailed solution for the above-mentioned questions is as follows

Since the given number ends with 5, so one's digit of its cube will also end with 5.

Q(v) Find the one’s digit of the cube of each of the following numbers.

1024

Answer: The solution to the above-mentioned question is as follows,

The given digit ends with 4. So the one’s digit of the cube of 1024 will be 4.

Q(vi) Find the one’s digit of the cube of each of the following numbers.

77

Answer: The detailed solution for the above-mentioned question is as follows,

The given number ends with 7, so its cube will end with 3.

Q(vii) Find the one’s digit of the cube of each of the following numbers.

5022

Answer: The detailed solution for the above-mentioned question is as follows,

Since the given number ends with 2, its cube will end with 8.

Q(viii) Find the one’s digit of the cube of each of the following numbers.

53

Answer: The detailed solution for the above-mentioned question is as follows,

Since the given number has 3 at units place, so, its cube will end with 7.

NCERT Solutions for Class 8 Maths Chapter 6 Cubes and Cube Roots: Topic 6.2.1
Subtopic Some Interesting Patterns

Q(a) Express the following numbers as the sum of odd numbers using the above pattern?

$6^{3}$

Answer: The detailed solution for the above-mentioned question is as follows,

$6^3 =$ 216 => 31 + 33 + 35 + 37 + 39 + 41

Q(b) Express the following numbers as the sum of odd numbers using the above pattern?

$8^{3}$

Answer: The detailed solution for the above-mentioned question is as follows

$8^3 =$ 512 => 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71

Q(c) Express the following numbers as the sum of odd numbers using the above pattern?

$7^{3}$

Answer: The detailed solution for the above-mentioned question is as follows

73 = 43 + 45 + 47 + 49 + 51 + 53 + 55

Q(i) The detailed solution of the above-written question is as follows,

Using the above pattern

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

Find the value of the following

$7^{3}-6^{3}$

Answer: The value of the following question is:

$7^{3}-6^{3} =$ $7^{3}-6^{3} = 1 + 7\times6\times3 = 1 + 126 = 127$

Q(ii) The detailed solution for all the above-written questions is as follows

Using the above pattern

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

Find the value of the following.

$12^{3}-11^{3}$

Answer: $12^{3}-11^{3} = 1 + 12\times11\times3 = 1 + 396 = 397$

Q(iii). Consider the following pattern.

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

Using the above pattern, find the value of the following.

$20^{3}-19^{3}$

Answer: The detailed solution for the above-written question is mentioned below,

$20^{3}-19^{3} = 1 + 20\times19\times3 = 1 + 1140 = 1141$

Q(iv) Consider the following pattern.

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

Using the above pattern, find the value of the following.

$51^{3}-50^{3}$

Answer: The detailed solution for the above question is mentioned below

$51^{3}-50^{3} = 1 + 51\times50\times3 = 1 + 7650 = 7651$

NCERT Solutions for Class 8 Maths Chapter 6 Cubes and Cube Roots Topic 6.2.1
Subtopic Cubes and Their Prime Factors

Q. Which of the following are perfect cubes?

1. 400

2. 3375

3. 8000

4. 15625

5. 9000

6. 6859

7. 2025

8. 10648

Answer: We will find it by prime factorization whether they make a pair of three prime numbers or not.

(1) $400 = 2\times2\times2\times2\times5\times5$ . So not a perfect cube.

(2) $3375 = 3\times3\times3\times5\times5\times5$ . So it is a perfect cube.

(3) $8000 = 2\times2\times2\times2\times2\times2\times5\times5\times5$ . So it is a perfect cube.

(4) $15625 = 5\times5\times5\times5\times5\times5$ . So it is a perfect cube.

(5) $9000 = 2\times2\times2\times3\times3\times5\times5\times5$ . So it is not a perfect cube.

(6) $6859 = 19\times19\times19$ . So it is a perfect cube.

(7) $2025 = 3\times3\times3\times3\times5\times5$ . So it is not a perfect cube.

(8) $10648 = 2\times2\times2\times11\times11\times11$ . So it is a perfect cube.

NCERT Solutions for Class 8 Maths Chapter 6 Cubes and Cube Roots
Topic 6.2.2 Smallest Multiple That is a Perfect Cube

Q. Check which of the following are perfect cubes.

(i) 2700

(ii) 16000

(iii) 64000

(iv) 900

(v) 125000

(vi) 36000

(vii) 21600

(viii) 10,000

(ix) 27000000

(x) 1000.

What pattern do you observe in these perfect cubes?

Answer: The detailed solution for the above-written question is as follows

By prime factorization:

(i) $2700 = 2\times2\times3\times3\times3\times5\times5$ . So it is not a perfect cube.

(ii) $16000 = 2\times2\times2\times2\times2\times2\times2\times5\times5\times5$ . So it is not a perfect cube.

(iii) $64000 = 2\times2\times2\times2\times2\times2\times2\times2\times2\times5\times5\times5= 80\times80\times80$ . So it is a perfect cube.

(iv) $900 = 2\times2\times3\times3\times5\times5$ . So it is not a perfect cube.

(v) $125000 = 2\times2\times2\times5\times5\times5\times5\times5\times5$ . So it is a perfect cube.

(vi) $36000 = 2\times2\times2\times2\times2\times3\times3\times5\times5\times5$ . So it is not a perfect cube.

(vii) $21600 = 2\times2\times2\times2\times2\times3\times3\times3\times5\times5$ . So it is not a perfect cube.

(viii) $10000 = 2\times2\times2\times2\times5\times5\times5\times5$ . So it is not a perfect cube.

(ix) $27000000 = 2\times2\times2\times2\times2\times2\times3\times3\times3\times5\times5\times5\times5\times5\times5$ . So it is a perfect cube.

(x) $1000 = 2\times2\times2\times5\times5\times5$ . So it is a perfect cube.

We observe that the numbers above which are perfect cube have the number of zeros in multiple of 3.

NCERT Class 8 Solution Exercise: 6.1
Page number: 76, Total questions: 4

Q.1(i) Which of the following numbers are not perfect cubes?

216

Answer: The detailed solution for the above-written question is as follows

By prime factorization of 216 gives:

$216 = 2\times2\times2\times3\times3\times3$

Since prime numbers are present in pairs of three, so the given number is a perfect cube.

Q.1(ii) Which of the following numbers are not perfect cubes?

128

Answer: We have 128. By prime factorization we get,

$128 = 2\times2\times2\times2\times2\times2\times2$

Since the prime numbers are not in pairs of three, so the given number is not a perfect cube.

Q.1(iii) Which of the following numbers are not perfect cubes?

1000

Answer: The detailed solution for the above written is as follows

By prime factorization of 1000 we get :

$1000 = 2\times2\times2\times5\times5\times5$ .

So the given number is a perfect cube.

Q.1(iv) Which of the following numbers are not perfect cubes?

100

Answer: The detailed solution for the above-written question is as follows

By prime factorization of 100 :

$100 = 2\times2\times5\times5$ .

Since prime numbers are not in pair of three so given number is not a perfect cube.

Q.1(v) Which of the following numbers are not perfect cubes?

46656

Answer: We have 46656, by prime factorisation:

$46656 = 2\times2\times2\times2\times2\times2\times3\times3\times3\times3\times3\times3$ .

Since prime numbers are in group of three. So the given number is a perfect cube.

Q.2 Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243

(ii) 256

(iii) 72

(iv) 675

(v) 100

Answer: This can be found by knowing about the prime factors of the number.

(i) 243 : $3\times3\times3\times3\times3$ .

So it must be multiplied by 3.

(ii) 256 : $2\times2\times2\times2\times2\times2\times2\times2$

So the given number must be multiplied by 2 to make it a perfect cube.

(iii) 72 : $2\times2\times2\times3\times3$

So 72 must be multiplied by 3 to make it a perfect cube.

(iv) 675 : $3\times3\times3\times5\times5$

So it should be multiplied by 5.

(v) 100 : $2\times2\times5\times5$

So it should be multiplied by 10.

Q.3 Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

(i) 81

(ii) 128

(iii) 135

(iv) 192

(v) 704

Answer: By prime factorization of given numbers:

(i) 81: $3\times3\times3\times3$

So given number needs to be divided by 3 to get a perfect cube.

(ii) 128: $2\times2\times2\times2\times2\times2\times2$ .

So the given number needs to be divided by 2 to get a perfect cube.

(iii) 135: $3\times3\times3\times5$

So the given number needs to be divided by 5 to get a perfect cube.

(iv) 192: $2\times2\times2\times2\times2\times2\times3$

So the given number needs to be divided by 3 to get a perfect cube.

(v) 704: $2\times2\times2\times2\times2\times2\times11$

So the given number needs to be divided by 11 to get a perfect cube.

Q.4 Parikshit makes a cuboid of plasticine of sides $5\: cm,2\; cm,5\; cm$. How many such cuboids will he need to form a cube?

Answer: Volume of cuboid is $5\times2\times5 = 2\times5\times5$ $cm^3$

To make it a cube need to make this a perfect cube number.

So we need $2\times2\times5$ cuboids or 20 cuboids.

Q. State true or false:

for any integer $m,m^{2}< m^{3}.$ Why?

Answer: The detailed solution for the above-written question is as follows.

False.

$m^2 < m^3$

Or, $0 < m^3 - m^2$

Or, $m^3 - m^2>0$

Or $m^2\left ( m-1 \right )>0$

Now, put any number less than 1, we see that this relation doesn't hold.

So for m<1 this condition is not true.

NCERT Class 8 Solution Exercise: 6.2
Page number: 77, Total questions: 2

Q.1(i) Find the cube root of each of the following numbers by the prime factorisation method.

64

Answer: The detailed solution for the above-written question is as follows

Prime factorization of 64 gives :

$64 = 2\times2\times2\times2\times2\times2$

So its cube root is $2\times2$ = 4

Q.1(ii) Find the cube root of each of the following numbers by prime factorisation method.

512

Answer: By prime factorisation of 512:

$512 = 2\times2\times2\times2\times2\times2\times2\times2\times2$

So its cube root is $2\times2\times2 = 8$

Q.1(iii) Find the cube root of each of the following numbers by the prime factorisation method.

10648

Answer: The detailed solution for the above-written question is as follows

Prime factorization of 10648 gives :

$10648 = 2\times2\times2\times11\times11\times11$

So, its cube root is 22.

Q.1(iv) Find the cube root of each of the following numbers by the prime factorisation method.

27000

Answer: The detailed solution for the above-written question is as follows

By the prime factorization method, we get :

$27000 = 2\times2\times2\times3\times3\times3\times5\times5\times5$

So its cube root is 30.

Q.1(v) Find the cube root of each of the following numbers by the prime factorisation method.

15625

Answer: The detailed solution for the above-written question is as follows

By prime factorization:

$15625 = 5\times5\times5\times5\times5\times5$

So its cube root is 25.

Q.1(vi) Find the cube root of each of the following numbers by prime factorisation method.

13824

Answer: The detailed solution for the above-written question is as follows

By prime factorization:

$13824 = 2\times2\times2\times2\times2\times2\times2\times2\times2\times3\times3\times3$

So its cube root is 24.

Q.1(vii) Find the cube root of each of the following numbers by the prime factorisation method.

110592

Answer: The detailed solution for the above-written question is as follows

By prime factorization:

$110592 = 2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times3\times3\times3$

So its cube root is $2\times2\times2\times2\times3$ = 48.

Q.1(viii) Find the cube root of each of the following numbers by the prime factorisation method.

46656

Answer: The detailed solution for the above-written question is as follows

By prime factorization, we get :

$46656 = 2\times2\times2\times2\times2\times2\times3\times3\times3\times3\times3\times3$

So its cube root is $2\times2\times3\times3$ = 36.

Q.1(ix) Find the cube root of each of the following numbers by the prime factorisation method.

175616

Answer: The detailed solution for the above-written question is as follows

By prime factorization, we get :

$175616 = 2\times2\times2\times2\times2\times2\times2\times2\times2\times7\times7\times7$

So its cube root is $2\times2\times2\times7$ = 56.

Q.1(x) Find the cube root of each of the following numbers by prime factorisation method.

91125

Answer: The detailed solution for the above-written question is as follows

By prime factorization, we get :

$91125 = 3\times3\times3\times3\times3\times3\times5\times5\times5$

So its cube root is $3\times3\times5$ = 45.

Q2. State true or false.

(i) Cube of any odd number is even.

(ii) A perfect cube does not end with two zeros.

(iii) If the square of a number ends with 5, then its cube ends with 25.

(iv) There is no perfect cube which ends with 8

(v) The cube of a two-digit number may be a three-digit number.

(vi) The cube of a two-digit number may have seven or more digits.

(vii) The cube of a single-digit number may be a single-digit number.

Answer: (i) False. The cube of an odd number can never be even.

(ii) True. A perfect cube number ends with zeros multiples of three.

(iii) False. We can say only about the unit's place.

(iv) False. Cube of numbers which end with 2 end with 8.

(v) False. It can never be.

(vi) False. It can never be. It can be proved by taking examples.

(vii) True. e.g. 1,2

NCERT Solutions For Class 8 Maths - Chapter Wise

Importance of Solving NCERT Questions of Class 8 Maths Chapter 6

  • Solving these NCERT questions will help students understand the basic concepts of Cubes and Cube Roots easily.
  • Students can practice various types of questions, which will improve their problem-solving skills.
  • These NCERT exercises cover all the important topics and concepts so that students can be well-prepared for various exams.
  • By solving these NCERT problems, students will get to know about all the real-life applications of Cubes and Cube Roots.

NCERT Class 8 Maths Solutions: Subject Wise

Students can use the links below to prepare efficiently in other subjects as well as Mathematics.

NCERT Books and Syllabus for Class 8

The following links will give students access to the latest CBSE syllabus and some reference books.

Keep Working Hard and Happy Learning!

Frequently Asked Questions (FAQs)

1. What are the important topics of chapter Cubes and Cube Roots ?

Finding cubes and cubes roots for numbers containing up to 3 digits, estimating square roots and cube roots are two important topics of this chapter.

2. How does the NCERT solutions are helpful ?

NCERT solutions not only helpful for the students if they stuck while solving NCERT problems but also they will get conceptual clarity as these solutions are provided in a very detailed manner.

3. Does CBSE provide NCERT solution for class 8 ?

No, CBSE doesn't provide NCERT solutions for any class and subject.

4. Where can I find the complete solutions of NCERT for class 8 ?

Here you will get the detailed NCERT solutions for class 8 by clicking on the link.

5. Where can I find the complete solutions of NCERT for class 8 maths ?

Here you will get the detailed NCERT solutions for class 8 maths by clicking on the link.

6. How many chapters are there in the CBSE class 8 maths ?

There are 16 chapters starting from rational number to playing with numbers in the CBSE class 8 maths.

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