Once you start dealing with unknown values like x and y, you’ll be glad you understand this important multiplication property.
Author
Taylor Hartley
Expert Reviewer
Jill Padfield
Published: February 2024
Once you start dealing with unknown values like x and y, you’ll be glad you understand this important multiplication property.
Author
Taylor Hartley
Expert Reviewer
Jill Padfield
Published: February 2024
Once you start dealing with unknown values like x and y, you’ll be glad you understand this important multiplication property.
Author
Taylor Hartley
Expert reviewer
Jill Padfield
Published: February 2024
Key takeaways
Table of contents
When you see an equation with parentheses, try your best not to panic. Parentheses are just another way of saying multiply. But how do you multiply when parentheses are involved? So far you’ve probably been using the order of operations, or the series of steps telling you when to multiply, divide, add and subtract.
But now we can look at these kinds of problems in a different way. The distributive property will teach us how to multiply numbers by other groups of numbers written in parentheses—mainly addition and subtraction equations. By mastering the distributive property, we can give ourselves a simple and easy way to understand multiplication with parentheses.
The distributive property will make more sense if we see it in action.
Let’s say you have an equation: 4 (2 + 5)
Now, your first instinct might be to add the 2 and the 5 together, then multiply by 4, which gives you: 4 (7) = 28.
The distributive property says that 4 can be distributed (or multiplied) to each individual component within the parentheses, too. In other words: 4 (2+5) = 4(2) + 4(5)
Now, watch how this gives us the same answer as above.
4 x 2 = 8
4 x 5 = 20
20 + 8 = 28
You might be thinking that it’s easier to add the numbers in parentheses together first, then multiply to get the right answer. But the distributive property really comes in handy when you have an nknown value like x or y.
So, let’s say your equation looks like: 3 (4 +x)
You can’t add 4 to X and get a number, right? But you can distribute so that your equation reads: 3(4) + 3(x) = 12 + 3x.
You might also see the distributive property of multiplication referred to as “the distributive law,” but that’s just another fancy name for it. You can think of the property as a law: it’s always true and works like a rule.
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The distributive property allows us to multiply one number by the individual parts of another group (the numbers grouped together by parentheses) and then carry on with adding or subtracting. This might sound a little confusing, so we’ve written the formula out for you:
a(b + c) = ab + ac
Here, the (a) is distributed to the group inside the parentheses, or the sum of (b) and (c). By distributing our (a), we can then multiply it by (b) and (c) separately, then add those products together to get our answer.
The problems we’ve walked through already demonstrate how the distributive property over addition works. Basically, you multiply the factor outside the parentheses to each value inside the parentheses, then add those values together to get your answer.
The values inside the parentheses are known as addends, or the values being added together. By distributing the first factor to each addend, we can perform the multiplication first, then calculate the sum of both of our products.
Take a look below:
7(8 + 9) = 7(8) + 7(9)
In the equation above, we can see that multiplying 7 by 8 and 7 by 9 individually gives us new products that become new addends:
7 x 8 = 56
7 x 9 = 63
Then, we add our new addends together: 56 + 63 = 119
That’s equal to the answer we would have gotten if we’d added 8 + 9 (which equals 17) and then multiplied by 7 ( 7 x 17 = 119).
The distributive property over subtraction works almost the same as with addition, but involves subtracting instead of adding. The equation below is a good example:
13 (9 – 4) = (13 x 9) – (13 x 4)
Here, we multiply the 13 by the 9 and by the 4 separately, then subtract the product of 13 and 9 and the product of 13 and 4.
13 x 9 = 117
13 x 4 = 52
117 – 52 = 65
To make sure you understand how the distributive property works, let’s run through a couple of practice problems.
These problems will cover the main two ways you’ll see the distributive property in action: addition and subtraction. Trust us: these exercises will help you as you move into more complex mathematics like algebra!
Solve the expression: 4 x (2 + 3)
First, let’s prove how we would normally solve this equation if we were using BODMAS (a method that tells us we solve what’s in brackets first, then order of powers or roots, then divide, then multiply, then add, then subtract). So, we would solve what’s in brackets first, adding 2 and 3 together, then multiply that sum by the number outside the brackets.
So, 2 + 3 equals 5. Now our expression looks like this: 4(5), or 4 x 5.
We know that 4 x 5 = 20, so that’s our final answer when solving this equation using the BODMAS method.
Now, let’s solve this same equation using the distributive property. The distributive property of multiplication says that when you multiply a number by a sum of two or more numbers, you can multiply the number outside the parentheses by each of the numbers inside, then add the results together. If we distribute the outside number to both of the inside ones, it should give us the following equation:
(4 x 2) + (4 x 3)
Now we’re ready to perform our multiplication. First, we have to solve (4 x 2), which equals 8. Next, we’ll solve (4 x 3), which gives us a product of 12. Since we’ve solved both our multiplication problems, our equation should look like this:
8 + 12
After distributing the 4 to both the 2 and the 3 inside the parentheses, we were able to find the sum of both of those products (which were 8 and 12). 12 + 8 = 20, proving that the distributive property gives us the same result as the classic BODMAS method.
Remember, the distributive property helps us break down a multiplication problem with addition inside the parentheses. We can multiply each number inside the parentheses by the number outside and then add the results together. It can make solving these problems easier, especially when there’s an unknown value inside the parentheses!
Let’s go ahead and take a look at how the distributive property works with subtraction. The basic rules are exactly the same, but make sure you remember what’s being subtracted from what. It’s important to keep the order of things being subtracted the same once you’ve distributed your first factor. Take a look below:
Solve the expression: 6 x (9 – 3)
Let’s start by solving this equation using order of operations. So, 9 – 3 equals 6. Now our expression looks like this: 6 x 6. We know from our multiplication tables that 6 x 6 = 36, so 36 is our final result.
The distributive property tells us that when multiplying a number by a difference of two other numbers, you can multiply the number outside the parentheses by each of the numbers inside. You can then subtract the results. For our problem, that should look like the following:
(6 x 9) – (6 x 3)
Remember what we said about keeping the order the same? In our new equation above, we’re still subtracting the product of 6 and 9 by the product of 6 and 3. If we were to change this order we would get an incorrect answer. A good rule of thumb is to follow the same order of subtraction that appeared in the original problem: (9 – 3).
(6 x 9) = 54, and (6 x 3) = 18, so now our equation looks like this:
54 – 18
Finally, all we need to do is solve our subtraction problem to get the final answer. Here, 54 – 18 = 36, the same answer we got from our first step. This proves that the distributive property over subtraction works as long as we follow the right steps.
There are a few other important multiplication properties you’ll need to remember going forward. The distributive property is one of the most important to master, but take a look at the other four to make sure you have a basic understanding of them, too.
The commutative property of multiplication talks about the placement of numbers in a multiplication equation, and how this placement can affect the product. Basically, the commutative property states that the order of numbers can be switched within an equation without changing the outcome of the multiplication. For example:
6 x 10 = 10 x 6
A x B = B x A
In the equation above, even if the 6 and 10 are switched in their positions, the outcome of the multiplication is the same. Swapping the positions of numbers does not affect the product.
The associative property states that when you multiply different numbers together, the grouping of these numbers has no effect on the product. You can group numbers together in different ways within a multiplication problem without producing a different value. Look below:
(15 x 3) 9 = 15 (3 x 9)
Here, the grouping of 15 and 3 on the left side is different from the grouping of 3 and 9 on the right. The associative property tells us that this doesn’t matter, as the product of the multiplication is the same regardless of how you choose to group your numbers.
The identity property of multiplication is a pretty basic rule that tells us that whenever we multiply a number by 1, the value of that number stays the same. All numbers operate the same when multiplied by 1—their value stays exactly the same. Follow along:
16 x 1 = 16
(a) x 1 = (a), where a represents any number.
The property of zero is about as simple as the identity property. The zero property states that any number (rational, whole or any kind of integer) multiplied by zero will equal zero. For example:
31 x 0 = 0
0 x (a) = 0, where (a) is any number.
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The basic idea behind the distributive property is that you multiply the term outside the parentheses by each of the inside terms. Once you’ve distributed the outside factor to both inside numbers, you can then add or subtract them. Remember, it’s important to keep the order of the numbers the same when subtracting, or you’ll get a different answer.
The distributive property of multiplication allows us to simplify complicated multiplication problems. It’s an efficient way of solving problems where one factor is multiplied by the sum or difference of others. When the contents inside the parentheses can’t be simplified any further, the distributive property helps us separate the two inside numbers, apply our multiplication rules to them, and then solve.
The distributive property of division follows similar concepts as the distributive property of multiplication. When used with division, the distributive property can help us understand how to break down larger numbers into smaller ones. This can help when dividing larger numbers, as the distributive property helps us break down those larger numbers into smaller factors. Those smaller factors are easy to calculate.
Yes, the distributive property works even when the outside factor is negative. All you need to remember is that when you distribute the outside number to both numbers on the inside, it will change both of their signs as well.
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Lesson credits
Taylor Hartley
Taylor Hartley is an author and an English teacher. When she's not writing, you can find her on the rowing machine or lost in a good novel.
Jill Padfield
Jill Padfield has 7 years of experience teaching mathematics. She is currently working as a Business Analyst, working to improve services for Veterans while earning a masters degree in business administration.
Taylor Hartley
Taylor Hartley is an author and an English teacher. When she's not writing, you can find her on the rowing machine or lost in a good novel.
Jill Padfield
Jill Padfield has 7 years of experience teaching mathematics. She is currently working as a Business Analyst, working to improve services for Veterans while earning a masters degree in business administration.
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