Commutative, Distributive and Associative Properties

The Distributive Property for 7th Grade

Practice

Add a new subject

What is the distributive property?

$a \cdot (b + c) = ab + ac$

$(a+b)(c +d) =ac+ad+bc+bd$

The distributive property is a rule in mathematics that says that multiplying a number by the sum of two or more numbers will give us the same result as multiplying that number by the two numbers separately and then adding them together.

For example, 4x4 will give us the same result as (4x2) + (4x2).

How does this help us? Well, it allows us to distribute, or to split up a number into two or more smaller numbers that are easier to work with. When we're working with large numbers, or expressions with variables, the distributive property can save us time and a headache!

Using the distributive property, we can break down a number into two or more smaller numbers using addition or subtraction, giving us an expression that is easier to solve without changing its final value.

Here is the formula for the basic distributive property:

$Z \cdot (X + Y) = ZX + ZY$

$Z \cdot (X - Y) = ZX - ZY$

The distributive property: the extended version

At first, we learn to use the distributive property using expressions with only one pair of parentheses. After we feel comfortable, we can move on to the extended distributive property.

The extended distributive property is used to multiply two sets of parentheses, one by the other.

Join Over 30,000 Students Excelling in Math!

Endless Practice, Expert Guidance - Elevate Your Math Skills Today

Naturally, the distributive property is useful in school, but it's also helpful in your daily life! Splitting up a check at a restaurant? Planning a group trip? The distributive property can make your real-life calculating and planning less of a headache, and everyone will be impressed by how quickly you can get things done. From basic distribution, to complex algebraic equations, to real world dilemmas - the distributive property is a fundamental part of the math we use every day, and getting a good understanding of how and when to use it can help you go far!

The distributive property of multiplication

Let's say you have a multiplication exercise with numbers that are too large to calculate in your head.

For example: $532\times8$

By using the distributive property, we will be able to break it down into simpler terms to solve:

With division exercises, the concept is the same. Again, we will use the distributive property to break down large numbers, and make our work easier. Suppose we are asked to solve the following: $76:4$

First, we will take the large, awkward number and round up to the next integer that is a multiple of the divisor (the number that 76 is being divided by, which is 4).

In our example, the number closest to $76$ that is a multiple of $4$ is $80$.

So,

$76:4=\left(80-4\right):4$

$=$

$80:4-4:4$

$=$

$20-1=19$

And we get:

$76:4=19$

The extended distributive property

At first, we try to focus on simpler expressions that have only one pair of parentheses. After we have mastered these, we can move on to the extended distributive property. Now, we will start solving exercises that have more than one pair of parentheses.

For example:

$\left(7+2\right)\times\left(5+8\right)$

We will use the extended distributive property to simplify the exercise. How?

We multiply each of the terms in the first pair of parentheses by each of the terms in the second pair of parentheses: