Sum of Cubes
Factors the sum of two cubes.
Formula
Live Calculator
MembersVariables
Example
8 + 27 = 2³ + 3³ = (2+3)(4 − 6 + 9) = 5×7 = 35. ✓
How it works
This identity rewrites a sum of two perfect cubes as a product of two factors, which makes the expression easier to simplify, solve, or factor. It works because expanding (a+b)(a^2-ab+b^2) gives a^3 + a^2b - a^2b - ab^2 + ab^2 + b^3, and the middle terms cancel, leaving a^3 + b^3. In other words, it is a special factoring pattern that always holds for real numbers a and b.
When to use it
Use this when you see an expression like x^3 + y^3 and want to factor it completely. It is especially useful in algebraic simplification, solving polynomial equations, and reducing expressions in calculus or higher-level math.
Worked example, step by step
Factor 64 + 125. 64 + 125 = 4^3 + 5^3 Use the sum of cubes formula: a^3 + b^3 = (a+b)(a^2 - ab + b^2) So, (4+5)(4^2 - 4·5 + 5^2) = 9(16 - 20 + 25) = 9(21) = 189
Common mistakes to avoid
- Forgetting the middle term is negative: a^2 - ab + b^2, not a^2 + ab + b^2
- Using the formula for a difference of cubes instead of a sum of cubes
- Not recognizing that numbers like 64 and 125 are perfect cubes before factoring
Did You Know?
The sum and difference of cubes factor neatly — a fact exploited constantly in calculus and cryptography.
Share this formula
More in Algebra
View allQuadratic Formula
IntermediateFinds the roots of any quadratic equation ax² + bx + c = 0.
Discriminant
IntermediateDetermines the nature of the roots of a quadratic equation.
Binomial Theorem
IntermediateExpands any power of a binomial into a sum of terms.
Logarithm Product Rule
IntermediateThe logarithm of a product equals the sum of logarithms.
Logarithm Quotient Rule
IntermediateThe logarithm of a quotient equals the difference of logarithms.
Logarithm Power Rule
IntermediateThe logarithm of a power equals the exponent times the logarithm.