The distributive property, also referred to as the distributive law, is a property of real numbers that states that multiplication distributes over addition. This means that multiplying by a group of numbers being added together is the same as multiplying each of the numbers in the group separately, then adding the products together. It can be expressed as:
a × (b + c) = (a × b) + (a × c)
Show that 3(7 + 5) = 3 × 7 + 3 × 5.
3(7 + 4) = (3 × 7) + (3 × 5)
3(12) = 21 + 15
36 = 36
The distributive property can be visualized as follows.
The figure above depicts the distributive property, namely, ab + ac = a(b+c), where a, b, and c are positive numbers (the distributive property holds true for negative numbers as well, but the figure would look different).
Applications of the distributive property
One of the most basic uses of the distributive property is to simplify multiplication problems. For example, we may not know what 5 × 23 is or want to calculate it in our heads, but we could break it up into 5(10 + 10 + 3), which we know from the distributive property is equal to 5 × 10 + 5 × 10 + 5 × 3, and may be a relatively easier mental calculation:
50 + 50 + 15 = 115
The distributive property is also commonly used in algebra. In some cases, expressions involving multiplication of groups of numbers can be simplified to solve the problem. In others, factoring expressions can serve the same purpose. Understanding the distributive property, along with the many other properties of real numbers, allows us to effectively tackle solving algebraic equations.