We know that the word 'Cube' is used in geometry. A cube is a solid figure which has all its sides are equal.
If the side of a cube in the adjoining figure is ‘a’ units then its volume is given by
= a ⋅ a ⋅ a
= a^{3} cubic units.
Here a^{3} is called 'a cubed' or 'a raised to the power 3' or 'a to the power 3'.
Now, consider the number 1, 8, 27, 64, 125,.......
These are called perfect cubes or cube numbers.
Each of them is obtained when a number is multiplied by itself three times.
Examples :
1 ⋅ 1 ⋅ 1 = 1^{3}
2 ⋅ 2 ⋅ 2 = 2^{3}
3 ⋅ 3 ⋅ 3 = 3^{3}
5 ⋅ 5 ⋅ 5 = 5^{3}
The following are the cubes of numbers from 11 to 20.
From the above table we observe the following properties of cubes :
Property 1 :
For numbers with their unit’s digit as 1, their cubes also will have the unit’s digit as 1.
For example,
1^{3} = 1
11^{3} = 1331
21^{3} = 9261
31^{3} = 29791
Property 2 :
The cubes of the numbers with 1, 4, 5, 6, 9 and 0 as unit digits will have the same unit digits.
For example,
14^{3} = 2744
15^{3} = 3375
16^{3} = 4096
20^{3} = 8000
Property 3 :
The cube of numbers ending in unit digit 2 will have a unit digit 8 and the cube of the numbers ending in unit digit 8 will have a unit digit 2.
For example,
12^{3} = 1728
18^{3} = 5832
Property 4 :
The cube of the numbers with unit digits as 3 will have a unit digit 7 and the cube of numbers with unit digit 7 will have a unit digit 3.
For example,
13^{3} = 2197
27^{3} = 19683
Property 5 :
The cubes of even numbers are all even; and the cubes of odd numbers are all odd.
For example,
18^{3} = 5832 (even)
27^{3} = 19683 (odd)
Property 6 :
The sum of the cubes of first n natural numbers is equal to the square of their sum.
1^{3 }+ 2^{3} + 3^{3} + ....... + n^{3} = (1 + 2 + 3 + ..... + n)^{2}
For example,
1^{3 }+ 2^{3} + 3^{3} + 4^{3} = (1 + 2 + 3 + 4)^{2}
1 + 8 + 27 + 64 = 10^{2}
100 = 100
Problem 1 :
Is 64 a perfect cube ?
Solution :
Using prime factorization, we can write 64 as given below.
64 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2
64 = 2^{3} ⋅ 2^{3} = (2 ⋅ 2)^{3} = 4^{3}
Hence, 64 is a perfect cube.
Problem 2 :
Is 500 a perfect cube ?
Solution :
Using prime factorization, we can write 500 as given below.
500 = 2 ⋅ 2 ⋅ 5 ⋅ 5 ⋅ 5
There are three 5’s in the product but only two 2’s.
Hence, 500 is not a perfect cube.
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