Class 8 — Mathematics (NCERT)
Cubes and Cube Roots
Class 8
- ✓By the end of this lesson students will be able to define and calculate the cube of a number.
- ✓By the end of this lesson students will be able to identify perfect cubes.
- ✓By the end of this lesson students will be able to understand the concept of a cube root.
- ✓By the end of this lesson students will be able to find the cube root of a perfect cube using the prime factorisation method.
Key concepts
When a number is multiplied by itself three times, the product obtained is called its cube. For example, the cube of 2 is 2 × 2 × 2 = 8. We denote the cube of a number 'n' as n³.
A natural number is called a perfect cube (or a cube number) if it is the cube of some natural number. For instance, 1, 8, 27, 64, 125 are perfect cubes because they are the cubes of 1, 2, 3, 4, and 5 respectively. To check if a number is a perfect cube, we perform its prime factorisation. If all the prime factors can be grouped into triplets of identical factors, then the number is a perfect cube.
The cube root of a number is the number that, when multiplied by itself three times, gives the original number. It is the inverse operation of finding a cube. For example, since 2³ = 8, the cube root of 8 is 2. The symbol for cube root is '³√'. If n = a³, then ³√n = a.
This method is used to find the cube root of a perfect cube. The steps are as follows:\n1. Express the given number as a product of its prime factors.\n2. Group the identical prime factors in triplets.\n3. From each triplet, take out one factor.\n4. Multiply these factors to get the cube root of the given number.
Key facts to remember
- 1The cube of an even number is always an even number.
- 2The cube of an odd number is always an odd number.
- 3A number ending in 0, 1, 4, 5, 6, or 9 will have its cube ending in the same digit.
- 4A number ending in 2 will have its cube ending in 8.
- 5A number ending in 8 will have its cube ending in 2.
- 6A number ending in 3 will have its cube ending in 7.
- 7A number ending in 7 will have its cube ending in 3.
- 8The symbol for cube root is ³√.
Worked examples
Example 1
Find the cube of 9.
Answer
729
Remember that n³ means n multiplied by itself three times, not n multiplied by 3.
Example 2
Is 512 a perfect cube? If yes, find the number whose cube it is.
Answer
Yes, 512 is a perfect cube. It is the cube of 8.
Example 3
Find the cube root of 9261 by prime factorisation.
Answer
21
Always ensure all prime factors are grouped into triplets for a perfect cube. If any factor is left ungrouped, the number is not a perfect cube.
Common mistakes
- ✗Confusing cubes with squares (e.g., calculating 5² instead of 5³).
- ✗Making errors in prime factorisation, leading to incorrect triplets.
- ✗Not grouping factors in triplets when finding cube roots; sometimes students group in pairs (like for square roots).
- ✗Forgetting to take only one factor from each triplet; instead, multiplying all factors in the triplet.
- ✗Incorrectly calculating the product of the factors after grouping.
Exam tips
- ★Memorise the cubes of numbers from 1 to 10 (or even 1 to 20) for quick recall and to check your answers.
- ★Always show all steps of prime factorisation and grouping clearly in your answer for cube roots.
- ★Double-check your multiplication and division calculations to avoid arithmetic errors.
- ★Practice finding cubes and cube roots of various numbers to build speed and accuracy.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.