The distributive property of multiplication is a property of real numbers that shows how we can break apart multiplication problems into separate terms. The property states that an algebraic expression a(b + c) becomes ab + ac. In other words, the multiplication of a distributes to both variables inside the parentheses, b and c.
How the Order of Operations Relates
Our understanding of the distributive property comes from the order of operations, commonly known as PEMDAS. When we rewrite expressions to spread out the multiplier, we are actually doing the first step of PEMDAS, which is handling parenthesis.
Let's take 3(2+5) as an example. We could either compute the value (2 + 5) first, giving us:
Or, we could use the distributive property to simplify the expression into 3(2) + 3(5):
Both methods give us the same answer.
Note: While the commutative property tells us that a(b + c) is equivalent to (b + c)a, we usually have the multiplier in front.
Using the Distributive Property of Multiplication in Algebraic Expressions
Keep in mind that this distributive property holds true for both addition and subtraction when you are distributing across a parenthesis. Let’s use another expression, 4(7-2) as an example.
We could either compute the value (7-2) first, giving us:
Or, we could use the distributive property to simplify the expression into 4(7) - 4(2). Notice how we keep the same minus sign across the terms.
Practicing the Distributive Property of Multiplication
For mathematicians, the distributive property of multiplication is a basic but widely used property. It will show up in your common core curriculum, along with other theorems like the associative and commutative properties. It’s also quite common in factoring and useful when we want to simplify and combine like terms in a more complicated expression.
In summary, the distributive property of multiplication states that for a multiplicative expression, we can distribute the multiplication across the terms within a parenthesis:
a(b+c) distributes to ab+ac.
a(b-c) distributes to ab-ac.