What is the Cube of a Number?
As we already know, multiplying a number three times by itself results in the cube of that number. A number having an exponent of three is also known as the cube of that number. The cube of a number can be a positive or negative number. The cube of odd numbers is always an odd number and for even numbers, the cube is even. The formula to find the cube of a number is,
\( n^3~=~n~\times~n~\times~n \), where n can be any number.
For example, the cube of 4, \( 4^3~=~4~\times~4~\times~4~=~64 \).
What is a Cube Root?
The cube root of a number is a value that, when multiplied thrice, produces the original number. In simple words, the cube root of a number is obtained when a number is divided by the same number three times. The representation of cube root is ‘\( \sqrt[3]{n} \)‘ , in which n can be any number. The cube root of a number can be positive or negative.
For example, the cube root of 343, \( \sqrt[3]{343}~=~\sqrt[3]{7~\times~7~\times~7}~=~7 \).
What is a Perfect Cube?
A perfect cube is a number that results in a cube root that is either a positive natural number or a negative number.
For example, 528 is a perfect cube, \( \sqrt[3]{528}~=~\sqrt[3]{8~\times~8~\times~8}~=~8 \).
Table of Cubes from 1 to 10
Number | Cube |
|---|---|
1 | \(1^3~=~1~\times~1~\times~1~=~1 \) |
2 | \(2^3~=~2~\times~2~\times~2~=~8 \) |
3 | \(3^3~=~3~\times~3~\times~3~=~27 \) |
4 | \(4^3~=~4~\times~4~\times~4~=~64 \) |
5 | \(5^3~=~5~\times~5~\times~5~=~125 \) |
6 | \(6^3~=~6~\times~6~\times~6~=~216 \) |
7 | \(7^3~=~7~\times~7~\times~7~=~343 \) |
8 | \(8^3~=~8~\times~8~\times~8~=~528 \) |
9 | \(9^3~=~9~\times~9~\times~9~=~729 \) |
10 | \(10^3~=~10~\times~10~\times~1~=~1000 \) |
Table of Cube Roots of ‘- 1000’ to ‘-1’
Number | Cube |
|---|---|
- 1000 | \(\sqrt[3]{-~1000}~=~\sqrt[3]{(-~10)~\times~(-~10)~\times~(-~10)}~=~-~10 \) |
- 729 | \(\sqrt[3]{-~729}~=~\sqrt[3]{(-~9)~\times~(-~9)~\times~(-~9)}~=~-~9 \) |
- 528 | \(\sqrt[3]{-~529}~=~\sqrt[3]{(-~8)~\times~(-~8)~\times~(-~8)}~=~-~8 \) |
- 343 | \(\sqrt[3]{-~343}~=~\sqrt[3]{(-~7)~\times~(-~7)~\times~(-~7)}~=~-~7 \) |
- 216 | \(\sqrt[3]{-~216}~=~\sqrt[3]{(-~6)~\times~(-~6)~\times~(-~6)}~=~-~6 \) |
- 125 | \(\sqrt[3]{-~125}~=~\sqrt[3]{(-~5)~\times~(-~5)~\times~(-~5)}~=~-~5 \) |
- 64 | \(\sqrt[3]{-~64}~=~\sqrt[3]{(-~4)~\times~(-~4)~\times~(-~4)}~=~-~4 \) |
- 27 | \(\sqrt[3]{-~27}~=~\sqrt[3]{(-~3)~\times~(-~3)~\times~(-~3)}~=~-~3 \) |
- 8 | \(\sqrt[3]{-~8}~=~\sqrt[3]{(-~2)~\times~(-~2)~\times~(-~2)}~=~-~2 \) |
- 1 | \(\sqrt[3]{-~1}~=~\sqrt[3]{(-~1)~\times~(-~1)~\times~(-~1)}~=~-~1 \) |
Solved Examples
Example 1:
Find the cube of 22.
Solution:
Cube of 22
\(22^3~=~22~\times~22~\times~22~=~10648 \).
Example 2:
What is the cube root of 1728?
Solution:
Use factor tree as a method to find the prime factors of 1728.
\(\sqrt[3]{1728}~=~\sqrt[3]{2~\times~2~\times~2~\times~2~\times~2~\times~2~\times~3~\times~3~\times~3} \)
\(~~~~~~~~~~~=~\sqrt[3]{2~\times~2~\times~2}~\times~\sqrt[3]{2~\times~2~\times~2}~\times~\sqrt[3]{3~\times~3~\times~3} \)
\(~~~~~~~~~~=~2~\times~2~\times~3 \)
\(~~~~~~~~~~=~12 \)
Example 3:
State whether – 1056 is a perfect cube or not.
Solution:
Cube root of – 1056
We can start with the factor tree of -1056.
\(-~1056=~\sqrt[3]{-~(2~\times~2~\times~2~\times~2~\times~2~\times~3~\times~11)} \)
\(~~~~~~~~~~=~\sqrt[3]{-~(2~\times~2~\times~2)}~\times~\sqrt[3]{-~(2~\times~2~\times~3~\times~11)} \)
\(~~~~~~~~~~=~\sqrt[3]{-~(2~\times~2~\times~3~\times~11)} \) – This does not lead to a natural number or a negative integer.
Hence, – 1056 is not a perfect cube.
Example 4:
John wants to make an ice cube with some water. The ice cube has a side length of 4 cm. What is the amount of water required to make the ice cube? If twelve ice cubes are made, what is the total quantity of water that is the total quantity of water needed to make twelve ice cubes.
Solution:
The side length of the cube is 4 cm. The amount of water required to make an ice cube is the volume of that cube.
Volume of cube = length \(\times\) breadth \(\times\) height
= 4 \(\times\) 4 \(\times\) 4
= 64 \(cm^3\).
So, the required amount of water for one ice cube is 64 \(cm^3\).
In order to make 12 ice cubes,
12 \(\times\) 64 = 768 \(cm^3\)
Hence, 12 ice cubes require 768 \(cm^3\) or 768 cubic centimeters of water.
The cube of a number is a number that is obtained when a number is multiplied thrice by itself.
For example, cube of 2 is 2 × 2 × 2 = 8.
The cube root of a number is a value that, when multiplied three times, produces the original value.
For example, cube root of 125 is 5.
The square of a number is obtained when a number is multiplied twice by itself, but the cube of a number is a number that is obtained when a number is multiplied thrice by itself.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three edges meeting at each vertex.