Cube of Sum / Difference Calculator
This calculator allows you to find the cube of a sum (a + b)³ or the cube of a difference (a - b)³ with step-by-step explanations.
Understanding Cube of Sum and Difference
The cube of a sum or difference is an algebraic identity:
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a - b)³ = a³ - 3a²b + 3ab² - b³
It is used in algebra, expansion, factorization, and solving cubic equations.
Example: (2 + 3)³ → 2³ + 3×2²×3 + 3×2×3² + 3³ = 8 + 36 + 54 + 27 = 125
Example: (5 - 2)³ → 5³ - 3×5²×2 + 3×5×2² - 2³ = 125 - 150 + 60 - 8 = 27
How the Calculator Works
The calculator automatically:
Step 1: Takes input values a and b
Step 2: Selects operation: cube of sum or cube of difference
Step 3: Applies the algebraic formula
Step 4: Displays result with step-by-step expansion
Examples
1. a = 2, b = 3, (a + b)³ → 125
2. a = 5, b = 2, (a - b)³ → 27
3. a = -2, b = 4, (a + b)³ → 8
4. a = 3, b = 1, (a - b)³ → 8
Why This Formula Matters
Cube of sum/difference identities simplify algebraic expansions, help in solving cubic equations, and are widely used in mathematics and physics problem-solving.
Common Mistakes to Avoid
- Forgetting middle terms (3a²b and 3ab²)
- Misplacing signs for (a - b)³
- Calculating each step incorrectly
Important: (a + b)³ is not equal to a³ + b³ – the full expansion includes 3a²b + 3ab².
Correct: Expand using (a ± b)³ = a³ ± 3a²b + 3ab² ± b³.
Frequently Asked Questions
Can a and b be negative?
Yes. The formula works for positive and negative numbers.
Can I use decimals?
Yes. Decimal values are supported.
Does the order matter for (a - b)³?
Yes. (a - b)³ ≠ (b - a)³ in sign, but magnitude may be same.
Conclusion
The cube of sum and difference is a powerful algebraic tool. This calculator quickly expands (a + b)³ and (a - b)³ with step-by-step results for any numbers.