Distributive Property

The distributive property says that the action done on numbers, which are given in brackets, can be done for each number outside of the brackets. It is one of the math characteristics that is used the most. The commutative property and the associative property are the other two big ones.

It is easy to remember what the distributive rule is. There are a number of features in math that can help us simplify not only arithmetic calculations but also algebraic expressions. In this piece, you’ll find out what the distributive property is, how to use the formula, and how to solve some problems.

Definition of Distributive Property

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What is Distributive Property

The Distributive Property is a mathematical property that is used to multiply a single value by two or more values inside a set of parentheses. The distribution property says that when a factor is multiplied by the sum or addition of two terms, each number must be multiplied by the factor, and then the two numbers must be added together. Symbolically, this quality can be said to be:

(B+C) = (AB+AC)
Where A, B, and C each have their value.
Let’s look at a simple example: the number 2(4+3).
Since the binomial “4 + 3” is in the brackets, the order of operations says that you should figure out the value of “4 + 3” and then multiply it by 2. It gives you the answer “14.”

How does the distributive property work?

With the distributive principle, we can divide a number by a sum or difference. It means that we can multiply the number by each part of the sum or difference and then add or take away the products. It can help you simplify expressions and figure out how to solve problems.

Here’s an example of how the distributive property can be used:
2(3 + 4) = 2(3) + 2(4)

The left side of the equation is 2 times the total of 3 and 4. On the right side of the equation, 2 is multiplied by each part of the sum, and the results are added together.
2(3 + 4) = 6 + 8
Now that we’ve added 6 and 8, we can make the right side of the problem easier:
2(3 + 4) = 14
So, we’ve shown that 2(3+4) = 14, which is an example of how the distributive property works.

Here’s another illustration:
5(x + y) = 5x + 5y
This equation shows that the feature of distribution works with variables as well.
You can use the distributive principle to simplify expressions, solve equations, and solve problems in the real world. For example, we can simplify the following formula by using the distributive property:
3x^2 + 2x – 10

We can divide the 3×2 into the terms 2x and -10:
3×2 plus 2x minus 10 equals 3×2(1) plus 3×2(2x) plus 3×2(-10)
We can now make the expression easier to understand by getting rid of the parentheses and putting like words together:
3×2 plus 2x minus 10 is 3×2 plus 6×3 minus 30×2. 3×2 plus 6×3 minus 30×2 is 6×3 minus 27×2 plus 2x.
By using the distributive rule, we have now made the expression easier to understand.

The distributive property is a very useful way to solve math questions. It is important to know how the distributive property works so that you can use it to your advantage.

Distributive Property with Variables

Take a look at this example: 6(2+4x)

Since the two numbers in the brackets are not the same, they can’t be added together. It means that it can’t be made any simpler. We need a different way, and the Distributive Property can help us do that.

When Distributive Property is used,
(6× 2) + (6 × 4x)

The brackets are gone, and each term is now increased by 6.

Now, you can make it easier to multiply each term by itself.
12 + 24x
The distributive principle of multiplication lets you simplify expressions where you multiply a number by a sum or difference. This rule says that the product of the sum or difference of two numbers is the same as the sum or difference of the two numbers. In algebra, the distributive property can be used for two math tasks:

• Distributive Property of Multiplication
• Distributive Property of Division

Distributive Property of Multiplication

The distributive feature of multiplication can be shown by adding and taking away. That means that the operation is inside the bracket, so the addition or subtraction between the numbers inside the bracket will happen between the numbers inside the bracket. Let’s use the cases here to figure out what these properties mean.

Distributive Property of Multiplication Over Addition

When you multiply a value by a sum, the distributive rule of multiplication over addition comes into play. For example, you want to multiply 5 by the sum of 10 and 3.

Since the terms are the same, we usually add them up and then increase them by 5.

5(10 + 3) = 5(13) = 65

But the property says that you should first increase each addend by 5. It is called “distributing the 5,” and then you can add the goods.

Before you add, 5(10) and 5(3) will be multiplied together.

5(10) + 5(3) = 50 + 15 = 65

The outcome is the same as it was before.

You probably don’t even realize that you’re using this method.

The two ways are shown in the equations below. On the left, we have 10 and 3, which we multiply by 5. This expansion can be written differently by using the distributive rule on the right-hand side. We divide by 5, multiply by 5, and then add the results. In each case, you will see that the result is the same.

5(10 + 3) = 5(10) + 5(3)

5(13) = 50 + 15

65 = 65

Distributive Property of Multiplication Over Subtraction

Now, let’s look at an example of a feature of multiplication over subtraction that has to do with how it spreads out.

Let’s say we need to multiply 6 by the difference between 13 and 5, which is (13 – 5).
It can be done in two different ways.
Case 1: 6 × (13 – 5) = 6 × 8 = 48
Case 2: 6 × (13 – 5) = (6 × 13) – (6 × 5) = 78 – 30 = 48
No matter which way you do things, the result will be the same.

Using the distributive laws of addition and subtraction, expressions can be changed to fit different needs. You can add and multiply when you multiply a number by a sum. Also, you can multiply each part of the addition first, then add the results. It is also true for taking away. In every case, you move the outer multiplier to each value in the parentheses so that multiplication happens before addition or subtraction with every value.

Distributive Property Examples

Some examples are:

Numerical examples:

• 2(3+4)=2(3)+2(4)=6+8=14
• −5(6−7)=−5(6)+(−5)(−7)=−30+35=5

Algebraic examples:

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

Real-world examples:

• If you have two bags of apples and each bag has three apples, you have a total of two times three, which is six apples.
• If you buy three things that each cost $4, you will pay $12.
• The size of a rectangular garden that is 10 feet long and 6 feet wide is 10 x 6 = 60 square feet.

Distributive Property of Division

We can use the distributive rule to divide larger numbers by breaking them up into smaller parts.

Let’s look at an example:
Q: How do you divide 84 by 6?
84 can be written as 60 plus 24
And so,
(60 + 24) ÷ 6
Now, by doing a division process for each factor in the bracket, we get the following;
(60 ÷ 6) + (24 ÷ 6)
= 10 + 4
= 14

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