Cubes and Cube Roots
Overview
Cubes and Cube Roots is a chapter in Class 8 Mathematics that explores the concept of cubing a number and finding its cube root. A cube of a number is obtained by multiplying it three times by itself. The chapter covers perfect cubes, properties of cubes, methods to find cube roots through prime factorisation, and patterns related to cube numbers. Understanding these concepts builds a strong foundation for higher mathematics.
Typed Notes
What is a Cube?
When a number is multiplied by itself three times, the result is called its cube. For example, 3 × 3 × 3 = 27, so the cube of 3 is 27, written as 3³ = 27. Numbers like 1, 8, 27, 64, 125, 216 are called perfect cubes because they are cubes of natural numbers 1, 2, 3, 4, 5, and 6 respectively. The cube of a negative number is always negative; for example, (−2)³ = −8. Cubes of numbers ending in specific digits follow a predictable pattern.
Properties of Cubes
There are several interesting properties of cube numbers. The cube of an even number is always even, and the cube of an odd number is always odd. If a number has m digits, its cube has at most 3m digits. Cubes of numbers ending in 0 end in 000; ending in 1 end in 1; ending in 2 end in 8; ending in 3 end in 7; ending in 4 end in 4; ending in 5 end in 5; ending in 6 end in 6; ending in 7 end in 3; ending in 8 end in 2; ending in 9 end in 9. The sum of consecutive odd numbers also shows a cube pattern.
Cube Roots by Prime Factorisation
The cube root of a number is the value that, when cubed, gives the original number. To find the cube root using prime factorisation, first express the number as a product of prime factors. Then group these prime factors into triplets of equal primes. Take one prime from each triplet and multiply them together to get the cube root. For example, to find ∛216: 216 = 2 × 2 × 2 × 3 × 3 × 3 = (2 × 3)³ = 6³, so ∛216 = 6. If any prime factor cannot form a complete triplet, the number is not a perfect cube.
Checking and Estimating Perfect Cubes
To check whether a number is a perfect cube, find its prime factors and see if every prime factor appears in groups of three (triplets). If yes, it is a perfect cube. For example, 500 = 2 × 2 × 5 × 5 × 5; here 2 appears only twice, so 500 is not a perfect cube. To make it a perfect cube, multiply by 2 to get 1000 = 10³. Similarly, to find the smallest number by which a given number must be divided to make it a perfect cube, identify the prime factors that do not form complete triplets.
Key Concepts
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- 01The cube of a number n is n × n × n, written as n³. For example, 5³ = 125.
- 02A perfect cube is a number that can be expressed as the cube of a whole number, e.g., 8 = 2³, 27 = 3³.
- 03The cube of an even number is always even, and the cube of an odd number is always odd.
- 04The cube root of a number x is the number whose cube gives x, written as ∛x. For example, ∛64 = 4.
- 05Prime factorisation method: group prime factors in triplets; one factor from each triplet gives the cube root.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
Find the cube of 12.
Is 1000 a perfect cube? Find its cube root.
Find the cube root of 13824 using prime factorisation.
What is the smallest number by which 72 must be multiplied to make it a perfect cube?
Find the cube root of −3375.
A cube of a number is obtained by multiplying it three times by itself (n³). Numbers whose cube roots are whole numbers are called perfect cubes. Cubes of even numbers are even and odd numbers are odd. The cube root (∛) is found using prime factorisation by grouping factors into triplets. These concepts help solve problems involving volumes, estimation, and number patterns in higher classes.
Chapter-end quiz
4 MCQsOption tap karo — turant pata chalega sahi ya galat, explanation ke saath.
Q1.What is the cube of 7?
Q2.Which of the following is a perfect cube?
Q3.What is ∛512?
Q4.The cube of an odd number is always:
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