The operation on numbers that are available in brackets can be distributed for each number outside the bracket, according to the distributive property. It is one of the mathematical properties that is used the most. Commutative and associative property are the other two important features.
The distributive property is straightforward to recall. Numerous mathematical properties allow us to make algebraic expressions as well as arithmetic calculations simpler. You will discover the definition of distributive property, its formula, and worked examples in this article.
What is Distributive Property?
The distributive property states that if a number is multiplied by the sum of two or more addends, the result is the same as if each addend is multiplied individually by the number, and then the products are added together.
Put differently, the expression A (B + C) can be simplified using the distributive property to AB + AC.
This property holds true for subtraction as well, and can be written as A (B – C) AB – AC. This means that the operand A is distributed across the other two operands.
Distributive Property Definition
The definition of the distributive property states that it enables us to take a factor and distribute it to every member or term of a group of things that have been added or subtracted. Rather than multiplying the factor by the group as a whole, we can distribute it to be multiplied by each member or term of the group separately.
Distributive Property Formula
The expression for the distributive property formula of a given value can be written as:
- a (b + c) = ab + ac
Distributive Property with Variables
It is impossible to simplify further since the two values included in the parentheses cannot be added because they are not like words. We require a different approach, and distributive property can be used in this situation.
Applying the Distributive Property
4× 2 + 4 × 4x
Every term is multiplied by 6 and the parenthesis is no longer present.
You may now make the multiplication of individual terms simpler.
8 + 16x
Expressions involving the multiplication of a number by a sum or difference can be made simpler thanks to the distributive principle of multiplication. The product of a sum or difference of a number is equal to the total or difference of the products, according to this property. In algebra, two arithmetic operations can have the distributive property as in:
- Distributive Property of Multiplication
- Distributive Property of Division
Here, we’ll go into more detail on the most popular distributive characteristic of multiplication over addition.
Distributive Property of Multiplication
Distributive Property of Multiplication can be represented using addition and subtraction. That indicates that an operation is carried out between the numbers enclosed in a bracket, such as addition or subtraction. Let’s use these examples to better comprehend these qualities.
Distributive Property of Multiplication over Addition
To multiply a number by the sum of two other numbers, we can utilize the distributive property of multiplication over addition. Here’s an example to help us understand this concept better:
Problem: Calculate the value of the expression 4(30+8) using the distributive property of multiplication over addition.
We can apply the distributive property to solve this expression. The number 4 is distributed across the two addends, which means we multiply each addend by 4 and then add the products. 4(30+8) = 430 + 48 = 120 + 32 = 152.
Distributive Property of Multiplication over Subtraction
The distributive property of multiplication over subtraction is similar to the distributive property of multiplication over addition, except for the operations involved which are subtraction and addition, respectively.
A(B − C) and AB − AC are equivalent expressions.
Example 1:
Simplify the expression 3(7 + 2) using the distributive property of multiplication over addition.
Solution: To simplify this expression, we can distribute the factor 3 across the parentheses, like this:
3(7 + 2) = 37 + 32 = 21 + 6 = 27
Therefore, the simplified expression is 27.
Example 2:
Simplify the expression 4(10 – 3) using the distributive property of multiplication over subtraction.
Solution:
To simplify this expression, we can distribute factor 4 across the parentheses, like this:
4(10 – 3) = 410 – 43 = 40 – 12 = 28
Therefore, the simplified expression is 28.
Distributive Property of Division
By splitting bigger numbers into smaller parts, we can divide them using the distributive property.
Here is an illustration to help:
Divide: 96 ÷ 8.
We can write 96 as 80+16.
Hence,
(80 + 16) ÷ 8
Now distributing division operation for each factor in the bracket we get;
(80 ÷ 8) + (16 ÷ 8)
= 10 + 2
= 12
Therefore, 96 ÷ 8 = 12.
Fun Facts:
The distributive law only applies to division when the dividend is decomposed or divided into partial dividends that are entirely divisible by the divisor, despite the fact that division is the opposite of multiplication.
Distributive Property Examples:
Example 1:
Solve the expression 7(8 + 9) by using the distributive property.
Solution:
Using the distributive property formula,
a × (b + c) = (a × b) + (a × c)
We will multiply the outside term by both the terms inside the brackets.
7(8 + 9)
= (7 × 8) + (7 × 9)
= 56 + 63
= 119
Therefore, the value of 7(8 + 9) = 119.
Example 2:
Solve 6(7 + 9) by using the distributive property formula.
Solution:
The distributive property formula is expressed as,
a × (b + c) = (a × b) + (a × c)
Now, let us multiply the outside term by both the terms inside the brackets.
= 6(7 + 9)
= (6 × 7) + (6 × 9)
= 42 + 54 = 96
Therefore, the solution of 6(7 + 9) is 96.
Example 3:
Solve using the distributive property formula: 7(8 + 14)
Solution:
The distributive property formula is expressed as,
a × (b + c) = (a × b) + (a × c)
Let us multiply the outside term by both the terms inside the parenthesis,
7(8 + 14)
= (7 × 8) + (7 × 14)
= 56 + 98
= 154
Therefore, the value of 7(8 + 14) = 154
The distributive property can be applied to division just like it can be applied to multiplication, but instead of combining terms, we break them apart. To do this with division, we need to divide the numerator into smaller amounts that can be evenly divided by the divisor.
Example: Let’s use the expression 24 ÷ 8 as an example. Using the distributive property, we can rewrite this as (16 + 8) ÷ 8. Then, we can break apart the numerator into 16 and 8, both of which are evenly divisible by 8, giving us (16 ÷ 8) + (8 ÷ 8) = 2 + 1 = 3. So, 24 ÷ 8 is equal to 3.
The distributive property states that if a number is multiplied by the sum of two or more addends, the result is equivalent to multiplying each addend by the same number and then adding the products together.
By distributing complex expressions into simpler terms, the distributive property simplifies problems, particularly those with multiple factors, and makes them easier to solve.
When utilizing the distributive property, each term inside the parentheses is multiplied by the outside factor, which eliminates the need for parentheses.
