Multiplication by Distributive Law

Introduction

The distributive law in mathematics, the algebraic operations that happen inside the bracket can be distributed to the number outside the bracket and get the same final answer. This is the most frequently used law in mathematics. It comes in handy while solving equations with variables. In this article, we will look more into this property and how it is applied.

Multiplication by Distributive Law

Distributive law is an algebraic property, it states, a value multiplied with a large sum of numbers can also be solved by multiplying the value with the numbers separately and taking the final sum.

Consider an algebraic equation, a (b + c)

Here, let d be the final sum of b + c. Here a can be multiplied with d and the final value can be obtained.

Or applying distributive law,

  a x b + a x c

The following equation can be solved, as shown above.

Distributive law helps simplify complex equations and makes it easier to solve them.

Let us take an example to understand distributive law:

Example:

  Consider,

                  5(2 + 4)

According to the order, the operations inside the bracket are performed first, and then operations outside the parentheses are followed. So, we add the values inside the parentheses, 2 + 4 which is 6, and multiply it with 5, which is 30. 

 Now distributive law states that the same problem can be performed by multiplying the value outside the parentheses with each number inside and adding them, the final answer is the same. 

 The above algebraic equation can be rewritten as,

     5(2 + 4) = 5 × 2 + 5 × 4 = 10 + 20 = 30.

 As shown in the above example, this property is most sought after. Distributive law is most frequently used as it comes in handy while solving equation which has variables. 

 Let us take an example for better understanding:

 Consider, 

                 5(2 + 3x)

The operation inside the parentheses cannot be solved or simplified. In such equations, the distributive property can be used to simplify them. Let us look at how to find multiplication by distributive property.

          5(2 + 3x) = 5 × 2 + 5 × 3x = 10 + 15x.

 Applying distributive law to such equations removes the parentheses and it becomes easier to simplify. It comes in handy, to solve higher-level equations with multiple variables that need to be simplified. 

 Distributive law can be used with two arithmetic operations, 

• Distributive property of multiplication
• Distributive property of division.

 This distributive property of multiplication allows us to simplify the equation by letting the value be multiplied over the sum or difference.

Distributive property of multiplication: 

The distributive property of multiplication can be done over addition or subtraction. That is, inside the parentheses over which the value is multiplied can either be addition or subtraction.

Let us look more into them along with examples.

 Over addition:

The distributive property of multiplication over addition can be expressed when the outside value is multiplied by the sum of numbers inside the parenthesis. For example, when you multiply 2 by the sum of 3 + 4.

As per the order of solving problems, first, the value is added and then multiplied.

2 (3 + 4) = 2 (7) = 14.

As per the distributive property the above equation can also be solved by multiplying 2 with the integers inside the bracket and then adding them.

2 (3) + 2 (4) = 6 + 8 = 14.

You can notice that both the methods give the same final answer.

Let us consider one more example for a clear understanding of this concept.

Consider 3 (4 + 2),

As per distributive law,

    3 (4 + 2) = 3(4) + 3(2)

    3 (6) = 12 + 6

     18 = 18

The left-hand side of the above equation is where the property is not distributed and the right-hand side shows the distributed equation.

Over subtraction:

 The distributive property of multiplication over subtraction can be expressed when the outside value is multiplied by the difference of numbers inside the parenthesis. For example, when you multiply 2 by the difference of 5 – 3.

As per the order of solving problems, first, the value is subtracted and then multiplied.

2 (5 – 3) = 2 (2) = 4.

As per the distributive property the above equation can also be solved by multiplying 2 with the integers inside the bracket and then subtracting them.

2 (5) – 2 (3) = 10 – 6 = 4.

You can notice that both the methods give the same final answer.

Conclusion

The distributive property of multiplication is important when it comes to solving complex equations with variables. It simplifies the equation and makes it easier to solve. There are more such properties called associative, commutative. All these properties serve similar purposes i.e., to simplify the equation and make them easier to solve.