Facts about Cubes
Cube of a number:
When a number is multiplied three times the product obtained is called the cube of the number.For a given number 'a', we define the cube as a × a × a, denoted by a3.
Perfect cube:
A natural number 'n' is said to be a perfect cube if (n = m3). It is the cube of another natural number.
For example
1 = 13
8 = 23
27 = 33
64 = 43
125 = 53
1, 8, 27, 64, 125 are perfect cubes.
A given natural number is a perfect cube if it can be expressed as the product of triplets of equal factors.
Cubes of a negative integer
The cube of a negative integer is always negative.
For example
(-1)3 = (-1) × (-1) × (-1) = -1
(-2)3 = (-2) × (-2) × (-2) = -8
(-3)3 = (-3) × (-3) × (-3) = -27
Cube of a rational number
We have (a/b)3 = a/b × a/b × a/b = (a × a × a)/(b × b × b) = a3/b3
Hence (a/b)3 = a3/ b3
For example
(i) (2/5)3 = 23/53 = (2 × 2× 2)/(5 × 5 × 5) = 8/125
(ii) (-2/3)3 = (-2)3/ 33 = {(-2) × (-2) × (-2)}/(3 × 3 × 3) = -8/27
Properties of cubes of numbers
The cube of every even natural number is even.
The cube of every odd natural number is odd.
The cube of a positive number is positive.
The cube of a negative number is negative.
Now let us see how to find the cube of each of these easily.First let us see a negative and zero ending number
(-60)3 = (- 6 x 10)3 here we know 63 and 103 so we can split the number and proceed the problem easily.
(-6) × (-6) × (-6) x (10 x 10 x 10)= -216000
Next consider a decimal number (4.5)3
This can be written as 45 / 10
(45/10)3 = (9 × 9 × 9)/(2× 2× 2)= 729/8 = 91 1/8
To get the decimal answer to the same problem, we can find the cube of 45 and then by dividing by 1000 which means shifting 3 decimal points from the last, we get the decimal answer.
453 = 91125 / 1000 = 91.125 is the answer.
Now consider (0.25)3 This can be first written as a decimal
(0.25)3 = (25/100)3= (1/4)3= 13/(4)3= 1 / 64
Tip of the week: Always make your problems simpler