If you multiply a number by itself and then by itself again (thrice), the product is a cube number. It is also called as a perfect cube. That is, if \(a\) is a number, its cube is represented by \(a^3\).
Example:
Let us find the cube number of \(3\).
Here, \(a = 3\).
\(a^3 = 3^3\)
\(= 3 \times 3 \times 3 = 27\)
Therefore, \(27\) is the cube number of \(3\).
The following table consist of cube numbers of the first ten numbers.
|
Number
|
Cube number
|
Number
|
Cube number
|
|
1
|
\(1^3 = 1\)
|
11
|
\(11^3 = 1331\)
|
|
2
|
\(2^3 = 8\)
|
12
|
\(12^3 = 1728\)
|
|
3
|
\(3^3 = 27\)
|
13
|
\(13^3 = 2197\)
|
|
4
|
\(4^3 = 64\)
|
14
|
\(14^3 = 2744\)
|
|
5
|
\(5^3 = 125\)
|
15
|
\(15^3 = 3375\)
|
|
6
|
\(6^3 = 216\)
|
16
|
\(16^3 = 4096\)
|
|
7
|
\(7^3 = 343\)
|
17
|
\(17^3 = 4913\)
|
|
8
|
\(8^3 = 512\)
|
18
|
\(18^3 = 5832\)
|
|
9
|
\(9^3 = 729\)
|
19
|
\(19^3 = 6859\)
|
|
10
|
\(10^3 = 1000\)
|
20
|
\(20^3 = 8000\)
|
1. The cube of every even number is even.
\(2^3 = 8\), \(4^3 = 64\), \(6^3 = 216\), \(8^3 = 512\), ...
Here, \(8\), \(64\), \(216\) and \(512\) are all even numbers.
2. The cube of every odd number is odd.
\(1^3 = 1\), \(3^3 = 27\), \(5^3 = 125\), \(7^3 = 343\), ...
Here, \(1\), \(27\), \(2125\) and \(343\) are all odd numbers.
3. If a natural number ends at \(0\), \(1\), \(4\), \(5\), \(6\) or \(9\), its cube also ends with the same \(0\), \(1\), \(4\), \(5\), \(6\) or \(9\), respectively.
4. The sum of the cubes of first \(n\) natural numbers is equal to the square of their sum.
That is, \(1^3 + 2^3 + 3^3 + 4 ^3 + …. + n^3 = (1 + 2 + 3 + 4 + … + n)^2\)
The inverse operation of a cube is cube root. The symbol used to represent the cube root is \(\sqrt[3]{}\).
A cube root is a unique value that gives us the original number when we multiply itself by three times.
The cube root of \(a\) is denoted by \(\sqrt[3]{a}\) or \(a^{\frac{1}{3}}\).
Example:
Find the cube root of \(64\).
\(\sqrt[3]{64} = \sqrt[3]{4 \times 4 \times 4}\) \(= \sqrt[3]{4^3}\) \( = 4\)
Therefore, the cube root of \(64\) is \(4\).
Steps to find the cube root of a number through prime factorisation:
Step 1: Find the prime factorisation of the given number.
Step 2: Group the factors in pair of three numbers (triplet).
Step 3: If there are no factor leftover, then the given number is a perfect cube. Otherwise, it is not a perfect cube.
Step 4: Now, take one factor common from each pair and multiply them.
Step 5: The obtained product is a cube root of a given number.