Perfect Cube Calculator
Check, Find & Analyze Perfect Cubes
Check if a number is a perfect cube, find perfect cubes in a range, or generate a list of perfect cubes with detailed analysis.
Calculator Mode
Enter any integer (positive or negative)
What is a Perfect Cube?
A perfect cube is a number that can be expressed as n³ where n is an integer.
Examples: 1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³), -8 ((-2)³)...
Note: Unlike squares, cubes can be negative!
Quick Facts
First 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Negative cubes: -1, -8, -27, -64, -125...
Sum formula: 1³ + 2³ + ... + n³ = [n(n+1)/2]²
Pattern: (n+1)³ - n³ = 3n² + 3n + 1
Properties of Perfect Cubes
- Perfect cubes can be positive or negative (unlike squares)
- The cube root of a perfect cube is always an integer
- All prime factors have exponents divisible by 3 in prime factorization
- The difference between consecutive cubes: (n+1)³ - n³ = 3n² + 3n + 1
- Sum of first n cubes equals the square of the sum of first n integers
- Every odd perfect cube is odd, every even perfect cube is even
- Perfect cubes can end in any digit (0-9)
Testing Methods
1. Cube Root Method:
Calculate ∛n. If it's an integer, n is a perfect cube.
2. Prime Factorization:
If all prime factors have exponents divisible by 3, it's a perfect cube.
3. Digital Root:
Perfect cubes have digital roots of 1, 8, or 9 (for positive cubes).
4. Last Digit Pattern:
Cubes ending in 0,1,4,5,6,9 → cube root ends in same digit.
Cube Patterns & Formulas
Algebraic Identities:
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a - b)³ = a³ - 3a²b + 3ab² - b³
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
Sum Formulas:
- 1³ + 2³ + ... + n³ = [n(n+1)/2]²
- Sum of first n cubes = (Sum of first n numbers)²
- 1³ + 3³ + 5³ + ... = n²(2n² - 1)
- 2³ + 4³ + 6³ + ... = 2n²(n + 1)²
Applications:
- Volume calculations (cubes)
- 3D geometry problems
- Cubic equations
- Number theory
- Physics (cubic relationships)
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