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Perfect Cubes


Cube :-

The cube of a number is the number raised to the power 3.
How do you find the volume of a cube of side s? You multiply s by itself three times. Thus    
Volume of cube = s× s ×s = s3
When a number is multiplied by itself three times, we get the cube of the number. 

The cube of 2 = 23 = 2× 2 ×2 = 8
The cube of 5 = 53 = 5× 5 × 5 = 125
8 and 27 are natural numbers which are cubes of natural numbers 2 and 3 respectively. Such numbers are called cubic numbers or perfect cubes. 

 

A cubic number or perfect cube is a natural number which is the cube of some natural number.

 

The following numbers are perfect cubes.
8 (=23) 27 (=33) 64 (= 43)
125 (= 53)216 (=63)343(=73)


Test for Perfect Cubes:
To test if a number is a perfect cube, resolve it into its prime factors. If a number can be expressed as the product of triples of equal factors, then it is a perfect cube.

 

Example

Are the following numbers perfect cubes? 
(i) 512   (ii) 27,000 (iii) 500

Solution

(i) 512 = (2 ×2 × 2) ´ (2×2 × 2) ´ (2 × 2 × 2)
     = 8 ×8×8 = 83
    Thus, 512 is a perfect cube. 
 

(ii) 27,000 = 27 × 1000 = 3× 3 × 3 × 10 × 10 × 10
      27,000 = (3 × 10) ×(3 × 10) × (3 ×10)
                  = 30 × 30 × 30 = 303
     27,000 is a perfect cube.

(iii) 500 = 125 × 4 = 5 × 5 ×5 × 2 × 2 = 53 × 2 × 2.

Hence 500 cannot be expressed as a product of triples of equal factors.

Therefore it is not a perfect cube.
Notice that in some cases, it is easy to factorise a number mentally into its prime factor, e.g. 27,000 and 500 can easily be factorised mentally. In other cases, factorisation has to be done by prime factorisation method.

 

Example

What is the least number that 500 must be multiplied with to make it a perfect cube?

Solution

We have seen that 500 = (2 × 2) × (5 × 5 × 5)
(2 × 2) × (5 × 5 × 5) = (2 × 2 × 2) × (5 × 5 × 5) = 23 × 53

Hence, to make the factors into a product of triples of equal factors, 500 should be multiplied by 2.

 

Example

 What is the smallest number that

686 must be divided by, to make it a perfect cube?

Solution

Factorising 686, we get 686 = 2 ´ 7 ´ 7 ´ 7

The factor 7 is a triple, but 2 is alone.
Therefore if we divide by 2,  the number obtained will be a perfect cube.

686 must be divided by 2 to make it is a perfect cube.

 

Properties of Cubes of Numbers  
Cubes of Even and Odd Numbers
If an even number is multiplied by itself three times, the result is an even number.

 

Examples: 2 ´ 2 ´ 2 = 8, 6 ´ 6 ´ 6 = 216, 12 ´ 12 ´ 12 = 1728
Also when an odd number is multiplied by itself three times, the result is an odd number.

 

Examples:

3 ´ 3 ´ 3 = 27,7 ´ 7 ´ 7 = 343, 11 ´ 11´ 11 = 1331
The cube of an even natural number is even.  

The cube of an odd natural number is odd.  

Cube of a Negative Integer
Look at the products given below. 

 ( - 1)3 = (- 1) ´ (- 1) ´ (- 1) = (- 1)
 (- 5)3 = ( - 5) ´ (- 5) ´ (- 5) = (- 125)

- 10 )3 = (- 10) ´ (- 10) ´ (- 10) = (- 1000)

We conclude that

The cube of a negative integer is negative.

Cube of a Rational Number (For your knowledge)
 


The cube of a rational number is given by
 





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