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Cubes of Rational Numbers

Let us observe the following cubes of rational numbers:

8
2
2
(a)    3 = × 2 × 2 = 2 × 2 × 2 = 27 Note: If m be a rational number,
n
3

3
3
3
3
3
3 ××
3
3

m
m
n ≠ 0, then
=
× −3(
)
× −3(
−  3
−  3
−  3
(b)    5   3 =    5   ×  −  3  ×    5   = ( −3) 5 ×× 5 ) = −27   n   n 3


 5
5
125
Example 1: Find the cube of the following numbers.
4
(a) 5.5 (b)
13 3
4
Solution: (a) (5.5) = (5.5) × (5.5) × (5.5) = 166.375 (b)   4   = 4 × 4 × 4 = 4 × × 4 = 64
3
 13 13 13 13 13 13 13 2197
×
×
Perfect Cubes
A number is called a perfect cube or cube number if it can be expressed as the product of a
number multiplied by itself thrice.
For example: 1 = 1 × 1 × 1 = 1 , 8 = 2 × 2 × 2 = 2 , 27 = 3 × 3 × 3 = 3 , 64 = 4 × 4 × 4 = 4 , etc.
3
3
3
3
Therefore, 1, 8, 27 and 64 are perfect cubes of 1, 2, 3 and 4 respectively.
So, we can say that, a natural number ‘n’ is called a perfect cube if there Quick Check
3
exists a natural number ‘m’ such that n = m × m × m = m . Is 12 a perfect cube?
Observe the following prime factorisation of the numbers and their cubes :
Number Prime factors Prime factorisation of its cube
4 2 × 2 4 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2 × 2 3
3
3
6 2 × 3 6 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2 × 3 3
3
3
8 2 × 2 × 2 8 = 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 × 2 × 2 3
3
3
3
3
3
10 2 × 5 10 = 1000 = 2 × 2 × 2 × 5 × 5 × 5 = 2 × 5 3
Each prime factor of a number appears three times in the prime factorisation of its cube.
To Check Whether a Number is a Perfect Cube or Not
This property can be used to determine whether a given
number is a perfect cube or not. 1. Numbers 4, 6, 9 and 10
cannot be written as
A natural number can be expressed as the product of triplets Note: the cube of any natural
of equal factors, then it is a perfect cube. number.
2. There are only 10 perfect
For example, (a) 64 = 2 × 2 × 2 × 2 × 2 × 2 cube numbers from 1 to
So, 64 is a perfect cube. 1000.
(b) 108 = 2 × 2 × 3 × 3 × 3
Here 2 × 2 is not a triple, so 108 is not a perfect cube.
To check if a number is a perfect cube or not, proceed as explained ahead.
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