Calculate the cube of a difference: (a − b)³ = a³ − 3a²b + 3ab² − b³. Enter values for a and b to compute the expanded form and result.
(5 − 3)³ = 2³ = 8 Expanded: a³ = 125 −3a²b = −3 × 25 × 3 = −225 +3ab² = 3 × 5 × 9 = 135 −b³ = −27 Sum: 125 − 225 + 135 − 27 = 8
Formula: (a − b)³ = a³ − 3a²b + 3ab² − b³. This is a key algebra identity for expanding cubed binomial differences.
What is the Cube of a Difference?
The cube of a difference is an algebraic identity that expresses the expansion of (a − b)³. It is one of the most important formulas of short multiplication, widely used in algebra, calculus, and engineering .
(a − b)³ = a³ − 3a²b + 3ab² − b³
This identity is derived from the binomial theorem or by multiplying (a − b) × (a − b) × (a − b). It is valid for all real numbers and algebraic expressions .
It is frequently used for:
- Factoring cubic expressions — recognizing the pattern a³ − 3a²b + 3ab² − b³
- Simplifying algebraic expressions — expanding or factoring
- Solving equations — especially cubic equations
- Geometry — calculating volumes of boxes with side length differences
- Physics and engineering — in calculations involving cubic relationships
The identity is closely related to the cube of a sum: (a + b)³ = a³ + 3a²b + 3ab² + b³ .
How to Use This Calculator
- Enter value for a: Input the first number or expression.
- Enter value for b: Input the second number or expression.
- Calculate: Click “Calculate” or press Enter.
- Swap: Click “Swap a ↔ b” to exchange the two values.
- Reset: Use “Reset” to restore default values.
- Interpretation: The result shows the cube of the difference, the expanded form, and the step-by-step calculation. For example, (5 − 3)³ = 8, with expansion terms: 125 − 225 + 135 − 27 = 8.