Short Multiplication Formulas

Short multiplication formulas let you expand and factor expressions without multiplying term by term: the square and cube of a sum or difference, the difference of squares, and the sum and difference of cubes. This page also lists the basic laws of arithmetic — commutative, associative and distributive — that every algebra rule is built on.

12 formulas · Grades 5–7

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All 12 formulas on this page
  1. Commutative law of addition
  2. Commutative law of multiplication
  3. Distributive law
  4. Associative law of addition
  5. Associative law of multiplication
  6. Square of a sum
  7. Square of a difference
  8. Difference of squares
  9. Cube of a sum
  10. Cube of a difference
  11. Sum of cubes
  12. Difference of cubes

Commutative law of addition

Grade 5
a+b=b+aa + b = b + a

Swapping the addends does not change the sum.

Legend:
  • a — Term a
  • b — Term b

Commutative law of multiplication

Grade 5
a⋅b=b⋅aa \cdot b = b \cdot a

Swapping the factors does not change the product.

Legend:
  • a — Factor a
  • b — Factor b

Distributive law

Grade 5
a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c

Multiplying a number by the sum of two terms.

Legend:
  • a — Number a
  • b — Number b
  • c — Number c

Associative law of addition

Grade 5
(a+b)+c=a+(b+c)(a+b)+c=a+(b+c)

The result of addition does not depend on how the addends are grouped.

Legend:
  • a — Term a
  • b — Term b
  • c — Term c

Associative law of multiplication

Grade 5
(a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c)

The result of multiplication does not depend on how the factors are grouped.

Legend:
  • a — Factor a
  • b — Factor b
  • c — Factor c

Square of a sum

Grade 7
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Short-multiplication formula for the square of a sum.

Legend:
  • a — Number a
  • b — Number b

Square of a difference

Grade 7
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Short-multiplication formula for the square of a difference.

Legend:
  • a — Number a
  • b — Number b

Difference of squares

Grade 7
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Factoring the difference of two squares.

Legend:
  • a — Number a
  • b — Number b

Cube of a sum

Grade 7
(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

The cube of the sum of two expressions.

Legend:
  • a — Number a
  • b — Number b

Cube of a difference

Grade 7
(a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

The cube of the difference of two expressions.

Legend:
  • a — Number a
  • b — Number b

Sum of cubes

Grade 7
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring the sum of two cubes.

Legend:
  • a — Number a
  • b — Number b

Difference of cubes

Grade 7
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring the difference of two cubes.

Legend:
  • a — Number a
  • b — Number b

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