CBSE Class 8 Maths Chapter 7: Cube and Cube Roots NCERT Solutions

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 8 PDF

CBSE Class 8 Mathematics Chapter 7, Cube and Cube Roots, introduces students to the fascinating world of perfect cubes and their roots. This chapter delves into identifying perfect cube numbers through prime factorization, where students learn to group prime factors into triplets to confirm if a number is a perfect cube. The NCERT Solutions offer a clear, step-by-step methodology for this process, along with methods to find the cube root of any given number. Through illustrative examples and detailed explanations, this chapter aims to solidify students' understanding of these concepts, providing them with essential skills for mathematical problem-solving and reinforcing their foundation in number theory for upcoming examinations.

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Quick info

Board	CBSE
Class	Class 8
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 7: Cube and Cube Roots

Chapter summary

Chapter 7, 'Cube and Cube Roots,' for Class 8 Maths NCERT Solutions, delves into the properties of cubes and how to identify perfect cubes. The exercises focus on prime factorization to determine if a number can be expressed as a product of three equal factors. Students will practice finding cube roots and understanding the relationship between a number and its cube. This chapter is crucial for developing number sense and algebraic skills related to powers and roots.

Learning outcomes

Understand the definition of a perfect cube.
Identify perfect cubes using prime factorization.
Determine if a number is not a perfect cube.
Apply the concept of grouping prime factors in triplets.
Distinguish between cubes and non-cubes.

Topics covered

Paper topics

Perfect Cubes
Prime Factorization
Identifying Perfect Cubes
Cube Roots
Number Theory

Important topics

Identifying perfect cubes using prime factorization
Prime factorization method for cube roots
Understanding the triplet rule for prime factors

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Questions and Solutions

Question 1

Which of the following numbers are not perfect cubes:

216
128
1000
100
46656

Solution:

To determine if a number is a perfect cube, we examine its prime factorization. A number is a perfect cube if all its prime factors appear in groups of three (triplets).

216 The prime factorization of 216 is $2 \times 2 \times 2 \times 3 \times 3 \times 3$ . We can group these factors as $(2 \times 2 \times 2) \times (3 \times 3 \times 3)$ or $2^3 \times 3^3$ . Since all prime factors appear in triplets, 216 is a perfect cube.
128 The prime factorization of 128 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ . We can group these factors as $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times 2$ or $2^6 \times 2^1$ . There is one factor of 2 that does not form a triplet. Therefore, 128 is not a perfect cube.
1000 The prime factorization of 1000 is $2 \times 2 \times 2 \times 5 \times 5 \times 5$ . We can group these factors as $(2 \times 2 \times 2) \times (5 \times 5 \times 5)$ or $2^3 \times 5^3$ . Since all prime factors appear in triplets, 1000 is a perfect cube.
100 The prime factorization of 100 is $2 \times 2 \times 5 \times 5$ . We can group these factors as $(2 \times 2) \times (5 \times 5)$ or $2^2 \times 5^2$ . Neither the factor 2 nor the factor 5 appears in a triplet. Therefore, 100 is not a perfect cube.
46656 The prime factorization of 46656 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$ . We can group these factors as $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3)$ or $2^6 \times 3^6$ . Since all prime factors appear in triplets, 46656 is a perfect cube.

Conclusion: The numbers that are not perfect cubes are 128 and 100.

Common mistakes

Incorrectly grouping prime factors.
Misinterpreting the prime factorization results.
Confusing cubes with squares.
Errors in performing prime factorization.

Revision tips

Practice prime factorization for each number thoroughly.
Ensure factors are grouped in sets of three for perfect cubes.
Review the definition of a perfect cube before starting exercises.
Cross-check your answers by cubing the obtained roots.

Practice MCQs

Q1. Which of the following numbers is a perfect cube?

Q2. What is the prime factorization of 1000?

Q3. A number is a perfect cube if its prime factors appear in groups of:

Q4. Which number is NOT a perfect cube based on its prime factors?

Frequently asked questions

What is a perfect cube?

A perfect cube is a number that can be obtained by multiplying an integer by itself three times. For example, 27 is a perfect cube because 3 x 3 x 3 = 27.

How can prime factorization help identify perfect cubes?

To identify a perfect cube using prime factorization, you find the prime factors of the number. If each prime factor appears in groups of three (triplets), the number is a perfect cube.

What does it mean if a prime factor does not appear in a triplet?

If any prime factor of a number does not appear in a group of three in its prime factorization, then the number is not a perfect cube.

Are all numbers perfect cubes?

No, not all numbers are perfect cubes. For example, 100 is not a perfect cube because its prime factors (2 x 2 x 5 x 5) cannot be grouped into triplets.

How do these solutions help with exam revision?

These solutions provide clear, step-by-step explanations for each problem, reinforcing the concepts of cubes and cube roots, which is essential for exam revision and understanding.

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