Meta Description: Distributive property explains the distributive power of the multiplication operation over the Addition and Subtraction signs. See how to solve it.
The Distributive property is one of the most used properties in mathematics. While you may not know it, you probably use this property most of the time you carry out a multiplication involving a bracket. It is also known as the distributive property of multiplication because it generally explains exactly how the multiplication operation is given as ‘X’ or ‘.’ Can be distributed successfully over the operations of addition and subtraction.
The distributive property is very important for almost all mathematics stages that deal with calculations. It plays a critical role in everything Algebraic Structure, Complex numbers, matrices, fields, rings, and polynomials. In the practical aspect of life, the distributive property can also be essential in solving certain math-related problems much more easily.
This piece will consider all there is to know about the distributive property, with different examples as well.
Distributive Property
The distributive property of binary operations is a property that generalizes the distributive law and states that the multiplication operation can be distributed over the addition and subtraction operations.
i.e
a . (b + c) = a . b + a . c
or
a x (b + c) = a x b + a x c
or
a(b + c) = ab + ac
The above is known as the distributive property of the multiplication operation over the addition operation. It is also known as the Distributive Property of Addition
Also, the property allows
a . (b – c) = a . b – a . c
or
a x (b – c) = a x b – a x c
or
a(b – c) = ab – ac
The above is known as the distributive property of the multiplication operation over the subtraction operation. It is also known as the Distributive Property of Subtraction
What Is Distributive Property?
The distributive property of binary operations states that the multiplication operation can be distributed over the addition and subtraction operations for a normal algebraic field.
The formal Distributive Property Definition is given below
Given a set D with the elements a, b and c and the two binary operations x and +,
The operation ‘x’ is left-distributive over ‘+’ if
a . (b + c) = (a . b) + (a . c)
and,
The operation ‘x’ is right-distributive over ‘+’
(b + c) . a = (b . a) + (c . a)
If both distributivities can be achieved, then the operation ‘x’ can be said to be distributive over the operation ‘+’
As was stated in the previous heading, the distributivity of the multiplication operation over the addition operation is also consistent with the distributivity of the multiplication operation over the subtraction operation.
The next section will consider how to solve the Distributive property.
How Do You Solve Distributive Property?
Solving the Distributivity properties is actually very easy. Students only need to ensure that they follow the definition stated in the latter. They will not have any problem arriving at their needed answer when they do. The steps are the same as with the Distributive Property Calculator available online.
Step 1: Ensure that there are three elements available in the set being considered.
Step 2: Two of the elements must be expressed with any of the binary operations’ +’ or ‘-‘
Step 3: The last element should be associated with the two elements stated in step 2 on the basis of the multiplication operation’ x’ or ‘.’
Step 4: Perform the calculation as outlined in the definition.
That being said, the next section will consider some examples to drive home how the property works.
Example of Distributive Property
We will consider some distributive Property Examples to help you have a perfect understanding of how it works.
Example 1
Find whether the distributivity property holds for the element below
4(5 + 6) and (5 + 6)4
Solution
Clearly, we can only tell if the x is distributive over + by finding the right and left distributivity.
For the right distributivity,
4(5 + 6) = (4 x 5) + (4 x 6)
4(11) = (20) + (24)
44 = 44
Clearly, the right distributive property holds.
Now let’s see for the left distributive property
(5 + 6)4 = (5 x 4) + (6 x 4)
(11)4 = (20) + (24)
44 = 44
Also, the left distributive property holds.
Verdict
Since right distributivity (44) is consistent with left distributivity (44), then the multiplication operation is distributive over the addition sign for the above elements.
Example 2
We will Consider the same elements as in example 1, but this time with a different sign.
4(5 – 6) and (5 – 6)4
The aim is also to find if the operation x is distributive over the operation –
Solution
For the right distributivity,
4(5 – 6) = (4 x 5) – (4 x 6)
4(-1) = (20) – (24)
– 4 = – 4
Clearly, the right distributive property holds.
For the left distributive property
(5 – 6)4 = (5 x 4) – (6 x 4)
(- 1)4 = (20) – (24)
– 4 = – 4
Also, the left distributive property holds.
Verdict
It is obvious that the multiplication operation is also distributive over the subtraction operation based on the consistency in both the right and left distributives.
Example 3
Find whether the multiplication sign is distributive over the elements
6(3 + 7) and (3 + 6)7
Solution
6(3 + 7) = (6 x 3) + (6 x 7)
6(10) = (18) + (42)
60 = 60
The right distributive law holds
Now for the left distributive law
(3 + 6)7 = (3 x 7) + (6 x 7)
(9)7 = (21) + (42)
63 = 63
The left distributive law holds
Verdict
Despite the fact that the left and right distributive holds when considered individually, the case is not the same for the whole distributive condition.
The right distributivity was 60, while the left distributivity was 63. As such, the multiplication sign is not distributive over the addition sign for the three elements considered.
Conclusion
The Distributivity property is not exactly difficult and works well for brackets having multiplications and other signs like addition and subtraction. As this piece fully highlighted, solving problems with the distributivity property is almost a one-way traffic irrespective of the plane it is being employed on. It can perfectly work on complex numbers, matrices, rings, number theory and, more importantly, algebraic theory. You can continue to practice as much as you want to ensure that you understand the property properly.