The Distributive Property is a useful algebra property that simplifies multiplication. Before we define the Distributive property, let us first revisit Commutativity.
Commutativity is an algebra property that says that changing the order with which a computation is done does not change the end result. Commutativity applies mostly to addition and multiplication but can also be applied to division in some cases.
Mathematically we define commutativity as:
The Distributive Property allows for multiplication across parentheses, and also quick multiplication of large numbers by breaking them into sums of smaller numbers which are then easier to multiply.
Observe from the above equations that the term outside the parentheses distributes across all the terms inside the parentheses.
Operators inside the parentheses must be addition or subtraction only and operators outside the parentheses must be multiplication or division only for the distributive property to apply.
One common mistake is not distributing the outside term across all the terms inside the parentheses like in the example shown below
Another important fact to remember is that signs (+, -) also distribute equally across parentheses as in the following example:
You might notice that the above equation is the same as
The distributive property can also be applied to division:
Another interesting case is shown below:
This is the same as:
The outside term distributes evenly into the parentheses i.e. it multiplies both terms
The distributive property allows for these two numbers to be multiplied by breaking up the larger one into a sum of smaller ones and then applying the property as shown below
The above multiplications are relatively easier: