Cubes and Cube Roots Class 8 Notes
Cubes and Cube Roots: Cubes and Cube Roots are essential topics for understanding numbers in maths. A cube is formed when a number is multiplied by itself three times. A cube root finds the number that produces a cube when multiplied three times.
What is a Cube?
A cube is the result of multiplying a number by itself three times.
Mathematically: Cube of a number = number × number × number
For example:
- Cube of 2 = 2 × 2 × 2 = 8
- Cube of 3 = 3 × 3 × 3 = 27
These results—8, 27, and 64—are called cube numbers or perfect cubes.
Perfect Cubes or Cube Numbers
A perfect cube is a number that can be written as the cube of an integer.
For example:
- 27 is a perfect cube because 27 = 3 × 3 × 3.
- 125 is a perfect cube because 125 = 5 × 5 × 5.
Some common perfect cubes are: 1, 8, 27, 64, 125, 216, and 1000.
If a number cannot be written as the cube of an integer, it is not a perfect cube.
Example: 50 and 200 are not perfect cubes.
Properties of Perfect Cubes
Perfect cubes show certain fixed patterns and rules. A table has been provided with the properties of perfect cubes.
| Properties of Perfect Cubes | |
| Property | Example |
| Cube of an even number is always even | 2³ = 8, 4³ = 64 |
| Cube of an odd number is always odd | 3³ = 27, 5³ = 125 |
| Cube of a negative number is always negative | (-2)³ = -8, (-3)³ = -27 |
| Cube of a number ending with 0, 1, 4, 5, 6, or 9 ends with the same digit | 4³ = 64, 5³ = 125, 6³ = 216 |
| Cube of a number ending with 2, 3, 7, or 8 follows a fixed pattern | 2³ = 8, 3³ = 27, 7³ = 343, 8³ = 512 |
| Cube can be written as the sum of consecutive odd numbers | 3³ = 7 + 9 + 11 = 27 |
| Cube of any number is greater than the number itself (except 0 and 1) | 2³ = 8 (>2), 5³ = 125 (>5) |
What is a Cube Root?
The cube root of a number is the number which, when multiplied by itself three times, gives the original number.
Symbolically, the cube root of a number n is written as ∛n.
Examples:
- ∛8 = 2 is 2 × 2 × 2 = 8
- ∛27 = 3 is 3 × 3 × 3 = 27
- ∛125 = 5 is 5 × 5 × 5 = 125
Finding Cube Roots using Prime Factorisation
The prime factorisation method helps to find the cube root of perfect cubes.
Step 1: Write the given number as a product of its prime factors.
Step 2: Make groups of three identical factors.
Step 3: Take one factor from each group. The next step is to multiply them.
Step 4: The product gives the cube root.
Example: Find ∛216
Step 1: 216 = 2 × 2 × 2 × 3 × 3 × 3
Step 2: Group the factors: (2 × 2 × 2) and (3 × 3 × 3)
Step 3: Take one factor from each group: 2 × 3 = 6
Step 4: Therefore, ∛216 = 6.
