# Distributive Property

## Distributive Property

The distributive property of multiplication over addition (or subtraction) states that the product of a sum (or difference) is equal to the sum (or difference) of the products.

### Examples

7 × 36 = 7 × (30 + 6) = 7 × 30 + 7 × 6 = 210 + 42 = 252
7 × 36 = 7 × (40 – 4) = 7 × 40 – 7 × 4 = 280 – 28 = 252

### Property

The distributive property can simplify calculations.
It is especially helpful in simplifying mental math.

An operation denoted by ⊗ distributes over an operation denoted by ⊕ if, regardless of the numbers a, b and c, we have :
a ⊗ (bc) = (ab) ⊕ (ac). This property is called the distributive property.

Specifically, the operation is left-distributive if : a ⊗ (bc) = (ab) ⊕ (ac).
It is right-distributive if : (a ⊕ b) ⊗ c = (a ⊗ c) ⊕ (c).

### Examples

• For the set of real numbers, multiplication distributes over addition :
12 × (3 + 10) = (12 × 3) + (12 × 10) = 36 + 120 = 156
• For the set of real numbers, multiplication distributes over subtraction :
25 × (20 − 5) = (25 × 20) − (25 × 5) = 500 − 125 = 375
• In set theory, the following equalities are obtained for the subsets A, B and C of U :
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
this means that the intersection of sets is left-distributive over the union of sets. It can also be shown that the union of sets distributes over the intersection of sets :
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)