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# Distributive Law of Multiplication

The distributive law states that if we multiply a number by a group of numbers that are added/subtracted together, then it will generate the same result if we do each multiplication separately.

So the results of both ways are the same.

The distributive law is one of the most frequently used laws in discrete mathematics. There are two more important laws that are Associate law and Commutative Law. We can easily simplify the arithmetic calculations as well as the algebraic expression with the help of various types of properties that exit in Math. In this section, we will learn about one property, which is the Distributive property. It is very easy to remember the distributive law. We will learn the definition, formula, and various examples of distributed law.

## Definition of Distributive Law

The distributive law is a type of algebraic law. In distributive law, we multiply a single value with two or more than two values within a set of parenthesis. According to this law, when a factor is multiplied by the sum of two terms, it must be necessary to multiply each of two terms by the factor and then do the addition operation. In the symbolical form, the distributive law can be represented in the following way:

Here X, Y, and Z are used to indicate the three different values.

To understand this, we will take a simple example, i.e., 9 (5 + 1).

We can see that the binomial “5 + 1” is in parenthesis. The order of operation describes that we will first find out the value of 5 + 1, and after that, we will multiply it by 9. After multiplication, we will get the resultant value 54.

## Distributive Law with Variables

To understand this, we will take an example, 5(6 + 8x)

In this example, we can see that the parenthesis contains two values, which cannot be added because these values are not like other terms. Therefore, we cannot simplify it any further. To do this, we need a different type of method. At this time, we can use the distribution law. So if we apply the distribution law, then we will get the following:

Here we can see that the parenthesis no longer exists, and the number 5 is multiplied with every term.

Now we will simplify the multiplication for individual terms like this:

With the help of distribution law of multiplication, we are able to simplify expression where we multiply a number by the difference or sum. This law states that the multiplication of sum/difference of a number and the sum/difference of the products are equal to each other. In discrete mathematics, we can use distribution law for two arithmetic operations, which are described as follows:

- Distributive law of Division
- Distribution law of Multiplication

## Distributive Law of Multiplication

We can only use addition and subtraction to express the distribution law of multiplication. According to this property, the operation exists inside the bracket. That means inside the bracket and between the numbers, there must be an addition or subtraction symbol. Now we will understand addition and subtraction separately in the distribution law.

### 1. Distribution law of Multiplication over Addition

When there is a case in which we want to multiply a value by a sum, in this case, we will apply the property of multiplication over addition. **For example:** Here, we will multiply the number 7 by the sum of 9 + 4.

There are two cases to perform the distribution of multiplication over addition.

**Case 1:** In the first case, we will first add the numbers. After that, we will multiply the added number by 5. Thus,

**Case 2:** In the second case, we will first solve the bracket. That means we will first multiply every number by 5. It can be called distributing the 5, and after that, we will add the products.

So before we doing addition, we will perform the multiplication of 7(9) and 7(4) like this:

As we can see that the result of both ways are the same.

Both the methods are described deeply by the below equations. On the LHS (left-hand side), there are addition of numbers 9 and 4, and then it is multiplied by 5. The right-hand side expression is rewritten by applying the distribution in which we first distribute 5, and then multiply them by 5, and lastly, we will add the results. In the results, we will see that the results of both sides are the same.

### 2. Distributive Law of Multiplication over Subtraction

When there is a case in which we want to multiply a value by subtraction, in this case, we will apply the property of multiplication over subtraction. **For example**: Here, we will multiply the number 5 by the subtraction of 10 and 3, i.e., (10 – 3).

There are two cases to perform the distribution of multiplication over subtraction.

**Case 1:** In the first case, we will first subtract the numbers. After that, we will multiply the subtracted number by 7. Thus,

**Case 2:** In the second case, we will first solve the bracket. That means we will first multiply every number by 7. It can be called distributing the 7, and after that, we will subtract the products.

As we can see that the result of both the cases are the same.

We can also use the distributive law of addition and subtraction for the purpose of rewriting expressions for different purposes. When we have a case where we multiply a number with a sum, it means that we may first perform addition and then multiplication. We can also multiply each addend first, and after that, we can add the products. The same process can be applied for subtraction as well. In each and every case, we disturb the outer multiplier to every value contained by the parenthesis so that before addition or subtraction, multiplication can occur with every value.

### Distributive law of Division

The large number can divide into smaller factors by breaking those numbers with the help of distributive law. We will understand this by an example, which is described as follows:

**Example:** In this example, we have to divide 84 by 6.

**Solution:** The number 84 can write as 60 + 24.

Hence 84 / 6 can be write as

(60 + 24) / 6

Now we will apply the distribution division operation on each factor in the above bracket, and then we will get the following:

(60 / 6) + (24 / 6)

= 10 + 4

= 14

### Examples of Distribution Law:

There are various examples of distribution law, which are described as follows:

**Example 1:** In this example, we have to solve the expression 4 * (2x^{4} + 7x) with the help of distribution law.

**Solution:** According to the distributive law:

X * (Y + Z) = XY + XZ

After putting the values, we will get the following

4 * (2x^{4} + 7x) = 4 * 2x^{4} + 4 * 7x

= 8x^{4} + 28x

**Example 2:** In this example, we have to solve the expression 4 * (8xy + 12yx) with the help of distribution law.

**Solution:** According to the distributive law:

X * (Y + Z) = XY + XZ

After putting the values, we will get the following

4 * (8xy + 12yx) = 4 * 8xy + 4 * 12yx

= 32xy + 48 xy

= 80xy

**Example 3:** In this example, we have to solve the expression 5 * (20 – 8) with the help of distribution law.

**Solution:** According to the distributive law:

X * (Y – Z) = XY – XZ

After putting the values, we will get the following

5 * (20 – 8) = 5 * 20 – 5 * 8

= 100 – 40

= 60

**Example 4:** In this example, we have to solve the expression 6x * (x^{3} + y) with the help of distribution law.

**Solution:** According to the distributive law:

X * (Y + Z) = XY + XZ

After putting the values, we will get the following

6x * (x^{3} + y) = 6x * x^{3} + 6x * y

= 6x^{4} + 6xy

### Why not Division and Multiplication

We cannot perform distributive law of multiplication over division and multiplication. Now we will understand this with the help of some examples, which are described as follows:

**Distribution law of Multiplication over Multiplication**

Suppose there are three integers 5, 9, and 3. Now we will assume that we can apply the distributive law on multiplication. Thus, this condition must be satisfied:

We will solve this expression by first multiplying the numbers of the bracket and then multiplying the multiplied number by 5. Thus, we will get the following:

5 (4 * 8) = 5 * 32

= 160

Now we will first solve the bracket. That means we will multiply every number by 5. After that, we will perform the multiplication operation.

(5 * 4) * (5 * 8)

= 20 * 40

= 800

As we can see that both results are not the same.

Therefore, it is proved that the distributive law of multiplication cannot be applied to multiplication.

**Distribution law of Multiplication over Division**

Suppose there are three integers 5, 9, and 3. Now we will assume that we can apply distributive law to division. Thus, this condition must be satisfied:

We solve this expression by first dividing the numbers and then multiplying the divided number by 5. Thus, we will get the following:

5 (9 / 3) = 5 * 3

= 15

Now we will first solve the bracket. That means we will multiply every number by 5. After that, we will perform the division operation.

(5 * 9) / (5 * 3)

= 45 / 15

= 3

As we can see that both results are not the same.

Therefore, it is proved that the distributive law of multiplication over division and multiplication cannot be applied.