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

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

NCERT Solutions PDF Class 8 PDF

This chapter delves into the concepts of cubes and cube roots for Class 8 Mathematics, aligned with CBSE guidelines. The NCERT Solutions provide a step-by-step approach to understanding how to determine if a number is a perfect cube by examining its prime factorization. Students will learn to group prime factors in triplets to identify perfect cubes and recognize numbers that are not perfect cubes due to unpaired prime factors. The solutions also cover methods for finding the cube root of numbers. This resource is designed to help students grasp these fundamental concepts, solve problems accurately, and prepare effectively for their examinations by reinforcing their understanding through clear explanations and examples.

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Quick info

Board	CBSE
Class	Class 8
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 7

Chapter summary

Chapter 7, 'Cubes and Cube Roots,' for Class 8 Maths NCERT Solutions focuses on understanding the properties of cubes and how to find them. The exercises guide students through identifying perfect cubes by prime factorization, where factors must appear in groups of three. It also covers determining if a number is not a perfect cube and introduces the concept of finding the cube root. These solutions offer a clear, step-by-step method to master these concepts.

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.
Solve problems related to cubes and cube roots.

Topics covered

Paper topics

Cubes
Perfect Cubes
Prime Factorization
Identifying Perfect Cubes
Cube Roots
Grouping of Prime Factors

Important topics

Identifying perfect cubes using prime factorization
Understanding the triplet rule for prime factors
Distinguishing between perfect cubes and non-perfect cubes

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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 need to find its prime factorization and check if each prime factor appears in groups of three (triplets).

216 The prime factorization of 216 is $2 \times 2 \times 2 \times 3 \times 3 \times 3$ . Here, the prime factors 2 and 3 each appear in a group of three. Therefore, 216 is a perfect cube.
$216 = 2^3 \times 3^3 = (2 \times 3)^3 = 6^3$
128 The prime factorization of 128 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ . We can group the factors as $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times 2$ . Here, one factor of 2 is left over and does not form a complete triplet. Therefore, 128 is not a perfect cube.
1000 The prime factorization of 1000 is $2 \times 2 \times 2 \times 5 \times 5 \times 5$ . The prime factors 2 and 5 each appear in a group of three. Therefore, 1000 is a perfect cube.
$1000 = 2^3 \times 5^3 = (2 \times 5)^3 = 10^3$
100 The prime factorization of 100 is $2 \times 2 \times 5 \times 5$ . We can group the factors as $(2 \times 2) \times (5 \times 5)$ . Here, neither the factor 2 nor the factor 5 appears in a group of three. 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 the factors as $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (3 \times 3 \times 3) \times (3 \times 3 \times 3)$ . All prime factors appear in groups of three. Therefore, 46656 is a perfect cube.
$46656 = 2^6 \times 3^6 = (2^2)^3 \times (3^2)^3 = 4^3 \times 9^3 = (4 \times 9)^3 = 36^3$

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

Common mistakes

Incorrectly grouping prime factors.
Assuming a number is a perfect cube without complete factorization.
Errors in prime factorization itself.
Confusing cubes with squares.

Revision tips

Practice prime factorization for each number thoroughly.
Ensure all prime factors are grouped in sets of three for a number to be a perfect cube.
Review the examples provided to understand the logic behind identifying non-cubes.
Focus on the grouping of factors as the key to solving these problems.

Practice MCQs

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

Q2. To determine if a number is a perfect cube, its prime factors must appear in groups of:

Q3. Which number is NOT a perfect cube?

Q4. The prime factorization of 1000 is:

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 × 3 × 3 = 27.

How can prime factorization help identify a perfect cube?

To check if a number is a perfect cube using prime factorization, you find all its prime factors. If each prime factor appears in groups of three (triplets), the number is a perfect cube. If any factor does not form a complete triplet, it's not a perfect cube.

Which numbers in Exercise 7.1 are not perfect cubes?

According to the solutions, the numbers 128 and 100 are not perfect cubes because their prime factors do not appear in complete triplets.

Are 216 and 1000 perfect cubes?

Yes, both 216 (2x2x2x3x3x3) and 1000 (2x2x2x5x5x5) are perfect cubes as all their prime factors occur in groups of three.

How do these solutions help with exam preparation?

These solutions provide clear, step-by-step methods to solve problems related to cubes and cube roots, reinforcing the understanding of prime factorization and its application in identifying perfect cubes, which is crucial for exam success.

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