Learning Objectives
- Understand perfect cubes and their properties
- Learn patterns in cube numbers
- Find cube roots using prime factorisation
- Identify whether a given number is a perfect cube
Key Concepts
Perfect Cubes
A natural number n is a perfect cube if it can be expressed as m³ for some natural number m. Examples: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
The cube of a negative number is always negative: (-2)³ = -8.
Properties of Cubes
- Cubes of even numbers are even; cubes of odd numbers are odd.
- A perfect cube does not necessarily end in specific digits like perfect squares.
- The cube of a number ending in 0 ends in 000; ending in 1 gives cube ending in 1; ending in 2 gives 8; ending in 3 gives 7; ending in 4 gives 4; ending in 5 gives 5; ending in 6 gives 6; ending in 7 gives 3; ending in 8 gives 2; ending in 9 gives 9.
Interesting Patterns
Sum of Consecutive Odd Numbers: Each perfect cube can be expressed as a sum of consecutive odd numbers.
- 1³ = 1
- 2³ = 3 + 5 = 8
- 3³ = 7 + 9 + 11 = 27
- 4³ = 13 + 15 + 17 + 19 = 64
Checking for Perfect Cubes
To check if a number is a perfect cube, find its prime factorisation. If each prime factor appears a number of times that is a multiple of 3, the number is a perfect cube.
Example: 216 = 2³ × 3³ → each prime appears 3 times → 216 is a perfect cube. ∛216 = 2 × 3 = 6.
Finding Cube Roots
Prime Factorisation Method: Express the number as a product of prime factors. Group them in triples. Take one factor from each triple and multiply them to get the cube root.
Estimation Method: For large numbers, consider the last digit to guess the unit digit of the cube root, and the remaining digits to estimate the tens digit.
Summary
Perfect cubes are cubes of integers. They can be expressed as sums of consecutive odd numbers. To check if a number is a perfect cube, ensure every prime factor appears a multiple-of-3 times. Cube roots are found by prime factorisation or estimation.
Important Terms
- Perfect Cube: A number that is the cube of an integer (e.g., 27 = 3³)
- Cube Root: The number which when cubed gives the original number (∛27 = 3)
- Prime Factorisation: Breaking a number into a product of prime factors
Quick Revision
- Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
- Cube of even = even; cube of odd = odd
- Cube of negative number = negative
- Check perfect cube: all prime factors must appear in groups of three
- Cube root by prime factorisation: group primes in triples and pick one from each
- Each perfect cube = sum of consecutive odd numbers