Understanding the Distributive Property

Have you ever come across an equation that looks like this: 3 × (4 + 2) and wondered how to simplify it? That’s where the distributive property of multiplication comes in! This fundamental math concept helps us simplify expressions and make calculations easier. In this post, we’ll explore what the distributive property is, see how it works through clear examples, and learn how to use it to remove parentheses from expressions. Let’s dive in!

What is the Distributive Property?

The distributive property is a rule that connects addition (or subtraction) and multiplication. It states that when you multiply a number by a sum (or difference), you can “distribute” the multiplication across each term inside the parentheses.

For example, one of the most common forms is:

a × (b + c) = (a × b) + (a × c)

You can also apply it to subtraction:

a × (b - c) = (a × b) - (a × c)

This property allows us to remove the parentheses and simplify expressions more easily.

What is the Distributive Property of Multiplication?

When you encounter the term “distributive property of multiplication,” it describes a situation where multiplication distributes over addition or subtraction — meaning the multiplication occurs outside the parentheses, while addition or subtraction happens inside.

Why is it called “of multiplication”?

It’s referred to as “of multiplication” because the multiplier (the number outside the parentheses) applies to every term within the parentheses. For example, in 3 × (4 + 5), the 3 multiplies the entire sum inside. Using the distributive property, we can rewrite it as 3 × 4 + 3 × 5. This shows that the multiplication is “of” the entire group inside the parentheses, not just a single number.

How does this help?

• It can turn a more complicated expression into simpler pieces that are easier to manage (especially when variables are involved).
• It lets you remove parentheses by spreading the multiplication.
• It connects sums inside parentheses with multiplication outside, that way you can simplify or manipulate expressions more flexibly.

Distributive Property Examples

Example 1: Numbers only

Solve: 5 × (2 + 7):

• Using standard order of operations: first add inside: 2 + 7 = 9, so 5 × 9 = 45.
• Using the distributive property: 5 × (2+7) = (5×2) + (5×7) = 10 + 35 = 45

Example 2: Breaking a factor into a sum

Suppose we want to solve 8 × 7. Instead of directly doing it, we break “7” into “5 + 2”:

8 × 7 = 8 × (5+2) = (8×5) + (8×2) = 40 + 16 = 56

This approach uses the idea that one of the numbers is written as a sum, allowing the distributive property to be applied.

Example 3: Using subtraction

Solve: 6 × (20−5):

• Using the distributive property: 6 × (20−5) = (6×20) − (6×5) = 120 − 30 = 90.

Here, we’re distributing the multiplier (6) across both parts inside the parentheses.

Using the Distributive Property to Remove the Parentheses

One of the major uses of the distributive property is removing parentheses (also called expanding an expression). Let’s walk through the steps:

1. Identify an expression of the form  a(b+c) or a(b−c).
2. Multiply "a" by each term inside the parentheses.
3. Maintain the sign of each term (if it’s a subtraction inside, keep the minus sign).
4. Combine like terms if needed (especially when variables are involved).
5. Result: the parentheses are gone, and you’ve got separate terms added or subtracted.

For example: To solve 5(x+6):

• Step 1: Recognize a=5, b=x, c=6.
• Step 2: Multiply: 5 × x = 5x. And 5 × 6 = 30.
• Step 3: Because it’s addition inside: 5(x+6) = 5x+30.
• No parentheses remain.

Why is This Important?

As you move into algebra, expanding expressions like 3(a + 4) or (2x + 5)(x − 1) becomes essential. It’s a key skill for simplifying and solving equations, allowing you to remove parentheses, combine like terms, and isolate variables. It also reflects a deeper understanding of how multiplication and addition work together to form the structure of algebraic expressions.

Want to see these strategies in action?

Watch our full video on YouTube for a step-by-step walkthrough and more helpful tips to support your learning:

Algebra doesn’t have to be intimidating! At TurtlEd, we help students build a strong foundation in math, including essential skills like understanding and applying the distributive property of multiplication. Our approach makes abstract concepts clear and engaging, helping students feel confident. Contact us to learn how our effective tutoring strategies can help your student thrive in math.