Exercise 6.4 — Cubes and Cube Roots

Exercise 6.4 – Cubes, Perfect Cubes & the Prime Factorisation Method

Exercise 6.4 from Chapter 6, "Square Roots and Cube Roots," of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) introduces the idea of a cube of a number and teaches students how to use prime factorisation to test whether a number is a perfect cube. The exercise also covers practical applications — finding the smallest number to multiply or divide by to make a number a perfect cube, and counting unit cubes that fit inside a cuboid.

This exercise builds directly on the idea of square numbers covered in earlier exercises of this chapter, and the same prime factorisation skills will be reused throughout Class 8, 9, and 10 whenever you need to simplify roots or check divisibility.

What Is a Cube Number?

When a number is multiplied by itself three times, the result is called the cube of that number. If a number can be written as a product of three equal factors, it is called a perfect cube or cubic number.

This idea connects to a visual model using unit cubes. A cube made of side-length 1 unit uses 1×1×1 = 1 unit cube. A cube of side-length 2 units is built from 2×2×2 = 8 unit cubes, and a cube of side-length 3 units needs 3×3×3 = 27 unit cubes. This pattern — side length cubed equals the total number of unit cubes — is exactly what the formula n³ represents.

Cubes of Numbers from 1 to 20

Memorising this table makes recognising perfect cubes much faster during exams:

Units Digit Pattern of Cubes

Another useful rule is that the cube of an even number is always even, and the cube of an odd number is always odd. The units digit of a number's cube follows a predictable pattern, which is especially helpful for quickly estimating cube roots:

Exercise 6.4 – Solved Questions, Step by Step

Common Mistakes to Avoid in Exercise 6.4

• Confusing squares with cubes: Squaring a number multiplies it by itself twice (n × n), while cubing multiplies it three times (n × n × n). Always re-check which operation the question asks for.
• Stopping prime factorisation too early: Always continue dividing until the quotient becomes 1. An incomplete factor tree can make a perfect cube look like it isn't one.
• Mixing up "multiply" and "divide" cases: If every prime factor except one is already in a group of three and that one factor needs just one more to complete its group, multiply. If a factor stands completely alone with no way to complete a group of three sensibly, divide it out.
• Forgetting to group factors correctly: When writing the prime factorisation, group identical factors together (e.g., write 2×2×2×2×2×2 as two groups of three 2s) to clearly see whether each prime appears in multiples of three.
• Arithmetic slips in large cubes: For numbers like 21³ or 30³, calculate step by step (first n×n, then multiply the result by n again) instead of trying to do it all in one go.

What Exercise 6.4 Prepares You For

The prime factorisation skills practised here are essential for the next step in this chapter — finding cube roots of perfect cubes using the prime factorisation method, where each group of three identical prime factors contributes one factor to the cube root. This naturally follows the techniques used for finding square roots earlier in this chapter, since both methods rely on grouping prime factors.

These ideas also reappear in Class 9 and Class 10 when simplifying expressions involving exponents and real numbers, and when working with algebraic identities involving cubes such as (a + b)³ and (a − b)³. Students preparing for CBSE, Telangana, or Andhra Pradesh board exams should be comfortable both calculating cubes directly and using prime factorisation to test for perfect cubes, as both types of questions appear regularly in exams.