Page 55 - ICSE Math 8

4 Cubes and Cube Roots

Key Concepts

• Cube of a Natural Number • Cube Root of a Negative Perfect Cube
• Properties of Cubes of Natural Numbers • Cube Root of the Product of Integers
• Cube Roots • Cube Root of a Rational Number
• Cube Root of a Natural Number • Patterns in Perfect Cubes
• Finding Cube Root by Prime Factorization

We have studied about the solid figure cube which has three dimensions, viz., length, breadth and height,
that are all equal. The space occupied by a cube is given by a × a × a, (where, a is the length of a side). This
product is known as cube of a.
In this chapter, we will learn about some special operations that will help us to find cubes and cube roots of
numbers.

Cube of a Natural Number

The cube of a number is the number with 3 as its exponent, i.e., a number raised to the power 3.
3
3
Thus, if x is a number, then x is its cube, where x = x × x × x. Maths Info
3
For example, (i) 4 = (4 × 4 × 4) = 64, i.e., cube of 4 is 64.
The cube of positive number
Ê 2ˆ 3 2 2 2 8 2 8 is always positive and that of
(ii) Á ˜ = 5 ¥ 5 ¥ 5 = 125 , i.e., cube of 5 is 125 . a negative number is always
3¯
Ë
Perfect Cube negative.
Any number which is a product of three identical numbers is called a perfect cube.
3
3
3
For example, 1 = 1 × 1 × 1 = 1, 2 = 2 × 2 × 2 = 8, 3 = 3 × 3 × 3 = 27, etc.
Thus, 1, 8, 27, 64, 125, etc., are perfect cubes.
Procedure for checking if a given natural number is a perfect cube
Step 1: Express the given natural number as a product of its prime factors.
Step 2: Group the factors in triplets of equal factors.
Step 3: If no factor is left after step 2, then the given natural number is a perfect cube, otherwise not.

To find a natural number whose cube is the given number, take out one factor from each triplet and multiply
them.

Example 1: Which of the following numbers are perfect cubes? Also, find the numbers
whose cubes they represent. 2 128
64
2
(a) 128 (b) 3,375 2 32
Solution: (a) Writing 128 as a product of its prime factors, we get 2 16

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 2 8
2 4
Grouping them into groups of three, you can see that one
2 is left ungrouped. 2 2
1
So, 128 is not a perfect cube.

43