What Is The Order of Operations?- Definition, Rules and Examples

The order of operations is a set of rules that must be followed in a specific order when solving an equation. 

We all are well aware of the basic four mathematics operations: addition, multiplication, division, and subtraction. Since our childhood, we have constantly been practising problems related to these four operations. These operations are utilized, from adding the bill of your purchase in a mall to dividing a pizza into four equal pieces. But, until now, we have solved problems related to each operation individually.

What if we have an expression in which we have all four operations working together? Do we multiply first, or do we subtract first? It can be troublesome if you follow the wrong approach. This may lead to wrong answers. Therefore the order of operations came into action.

Analyzing any mathematical statement using arithmetic operations such as division, multiplication, addition, and subtraction is referred to as operations in mathematics. Let’s look at the laws of the order of operations and see how well we can recall them using simple strategies. In this article, you learn the order of operations, the order of operations rules, the mathematical order of operations, and solve order of operations problems. 

Why is the Order of Operations Implemented?

When evaluating equations in arithmetic, there may be numerous operations to perform, and simplification, in the end, provides varied outcomes. For each phrase, however, there can only be one correct response. We use principles to simplify any given mathematical expression to find the proper answer. These principles are based on all of the basic math operators.

For the sole purpose of always getting the correct answers for any mathematical equation, the order of operations was implemented.

What is the Order of Operations?

In Math, operators such as addition (+), subtraction (-), multiplication (×), and division (÷) are given priority over one another. We cannot solve an expression containing either of the 2 operators according to ourselves. We are ought to follow the order of operations rules which states that in any expression, either arithmetic or algebraic, the brackets will be evaluated first. The order will be calculated in the second place. Multiplication and division will be evaluated third and finally, addition and subtraction will be simplified. 

The priority of solving any expression is mentioned in the table below:

Priority	Operation
First	Brackets (), {}, [ ]
Second	Orders ( Exponents, Square root, Cube root, Logarithmic, etc.)
Third	Division or multiplication (Whichever operation is to the left side of the expression will be evaluated first)
Fourth	Addition or subtraction (Whichever operation is to the left side of the expression will be evaluated first)

Order of Operations Definition

As previously stated, the order of operations is a collection of basic precedence principles that we employ while solving any mathematical equation that involves many operations. When a subexpression exists between two operators, the operator that appears first in the table mentioned in the previous section shall be implemented.

Let us now understand the order of operations, rule by rule according to the set priority:

Order of Operations Rules

While applying the order of operations on any expression, the first thing you need to do is look carefully for different operators being applied to the expression. While solving any expression, always move from your left side to your right side. This will help you ease the simplification of that expression. Now read the rules mentioned below to grasp the concept of order of operations completely:

Rule 1: You need to always check for any brackets or parentheses in the expression. The most common brackets you will see in mathematical expressions are ‘()’, ‘{}’, and ‘[]’. When a term contains all three brackets, then solve the equation in the following way:

• Solve the terms inside the round bracket ‘()’ or parentheses first.
• Solve the terms inside the curly brackets ‘{}’ second.
• Last, solve the terms inside the square brackets ‘[ ]’.

Also, remember if there are multiple operations inside any one of the brackets, make sure to use the order of operations to solve them. 

Rule 2: After you’ve solved the numbers in the parenthesis, search for any terms following orders like exponentiated terms, root terms, logarithmic terms, trigonometric terms, etc., and solve them.

Rule 3: All left is placing the four basic operators in the right order. You can either do multiplication or division whichever operator is first from the left side of the expression.

Rule 4: The final step is to add or subtract the terms in the same left-to-right order.

If you remember these five rules then you have mastered the order of operation. No matter how long the expression is, you can easily solve it without hesitation. If you are not confident in remembering these rules, there are two easy methods to memorize them. They are known as BODMAS and PEMDAS. In the next sections, you will learn about these two acronyms for order of operations. 

Ways to Remember Order of Operations – BODMAS and PEMDAS

There exist two very important ways of remembering the order of operations. They are abbreviated as the PEMDAS rule and the BODMAS rule. The letters in these acronyms specify the mathematical operations. The letter which is present first is applied first on any equation.

Let us now learn what each of the letters in these acronyms stands for:

Order of Operations BODMAS

BODMAS is the most common acronym for the order of operations. Many of you must have heard of it somewhere in your life. When expanded, it stands for:

• B – Brackets ( ), { }, [ ]
• O – Order
• D – Division (÷)
• M – Multiplication (×)
• A – Addition (+)
• S – Subtraction (-)

Order of Operations PEMDAS

PEMDAS is also an acronym for the order of operation. Still, it is heard less in our day-to-day life, but it is equally as effective as BODMAS.

• P – Parentheses (), {}, [].
• E – Exponents (a2) (For example, here, a is a number with exponent 2)
• M – Multiplication (×)
• D – Division (÷)
• A – stands for Addition (+)
• S – stands for Subtraction (-)

This is the most effective method for remembering the sequence of events. The funny phrase “Please Excuse My Dear Aunt Sally” is a good way to memorize PEMDAS.

Both of these methods are used to simplify our understanding of the order of operations. We shall see some order of operations examples to clarify this topic.

What is the Order of Operations in Math? 

The order of operations is a rule that specifies the correct sequence of steps to evaluate a mathematical expression. If you have an expression with only one operation (for example, only addition, only subtraction, only multiplication, or only division), the correct way to solve it is from left to right. However, for expressions with multiple operations, the order of operations must be followed. 

Order of Operations Example

Let us look at the different examples mentioned below to understand the accuracy of the rules used in the order of operations.

1. For evaluating arithmetic operators

Expression 1: 2 + 3 x 5 

• Correct way: The correct way to solve this expression is by first multiplying 3 x 5 = 15 and then adding 2 to the result. 15 + 2 = 17.
• Incorrect way: If you add 2 + 3 first. We get 6 x 5 = 30, which is incorrect in solving this expression.

Expression 2: 15 ÷ 3 x 2 – 6

• Correct way: The right way to solve this expression is by first dividing 15 and 3 = 5 because it is closer to the left of the expression. Then multiply 5 by 2 = 10 and subtract 6 from this, which gives 10 – 6 = 4. 
• Incorrect way: If you multiply 3 x 2 = 6, then subtract 6 from it we get 15 ÷ 0 = not defined. This is the incorrect way to solve this expression.

2. For evaluating orders

Expression 1: 6 x 32

• Correct way: The correct way to solve this expression is by first solving the order, that is, 32 = 9. Now multiply this with 6. 9 x 6 = 54.
• Incorrect way: If you multiply 6 x 3 = 18, then square the answer we get, 182 = 324, which is incorrect. 

Expression 2: 2 x sin 30

• Correct way: First, evaluate the order sin 30 = ½. Now multiplying 2 with the answer. 2 x ½ = 1
• Incorrect way: If you multiply 30 by 2 = 60 and solve the order, we get sin 60 = 3/2. Which is an incorrect answer

3. For evaluating brackets

Expression 1: (12 + 3) x 4

Solution: In this expression, we can see that parentheses are present; hence we will solve the content inside the parentheses first. Therefore, 12 + 3 = 15. Now we multiply 15 by 4 = 60. 

Expression 2: 4 – 32 ÷ 8 

4 – 32 ÷ 8 = 8 – 8 = 0 is the solution. (This is correct. This is the proper way.) 

Consider another approach to the same expression. 

4 – 32 ÷ 8 = (-28) ÷ 8 = -3.5 (Incorrect (This is an incorrect method.) 

Expression 3: 8 x (6 + 6) 

8 x (12) = 96 (This is the correct way to solve the parentheses.) 

Consider another method for the same expression. 

8 x (6 + 6) = 48 + 6 = 54 (This is an incorrect way to solve the parentheses.) 

Note: We must observe the pattern of operators when executing the order of operations on any given phrase.

From the above-mentioned rules, we get to know how to operate multiple operations if they come in one equation. It is significant to learn these concepts so that you eliminate every possible error while doing calculations. Moreover, mastering the order of operations will help get more scores in the exams.