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# Order of Operations and Grouping Symbols

## Introduction

In simplifying mathematical expressions consisting of the same type of operation, we perform one operation at a time generally starring from the left towards the right. If an expression has more than one fundamental operation, you cannot perform operations in the order they appear. Some operations have to be performed before than the others.  This is each operation has its own precedence. Generally, the order in which we perform operations sequentially from left to right is division, multiplication, addition, subtraction. But when the expressions make use of brackets as well, we have a set of rules that defines the precedence of the operations. Let us learn the manner in which this rule is defined.

Let us understand this through an example.

Suppose we want to find the value of 2 + 3 x 5

There are two ways to find the value of the above expression.

In the first method, let us move from left to right, solving the expression in that order. This means, first we will find the sum of 2 and 3. We will get, 2 + 3 = 5.

Now we have 2 + 3 x 5 = 5  x 5

Now, we find the product of 5 with 5 to get the answer as 25. Therefore, we have,

2 + 3 x 5 = 25 ……………………………….. ( 1 )

Now, let us find the value of this expression through another method.

We will first find the value of 3 x 5 and add this product to 2. So, we get,

3 x 5 = 15 and 15 + 2 = 17

Hence, now we have,

2 + 3 x 5 = 17 ………………………… ( 2 )

From ( 1 ) and ( 2 ) we can see that both the results obtained are not the same. This validates the need for having precedence of operators so as to receive the same answer to the given algebraic expression.  This precedence is known as the order of operations.

## What is Order of Operations?

Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations. Without consistent order of operation, one can make big mistakes during computation. There are a number of rules that define the order of operations, depending upon the involvement of the operators, brackets exponents and other mathematical symbols for operations. This rule is known as BODMAS or PEMDAS. Let us understand the rule in further detail.

### BODMAS and PEMDAS

BODMAS is a sequence to perform operations in an arithmetic expression. It is an acronym where each alphabet represents a particular operation. BOMAS stands for –

B – Brackets

O – Orders (powers/indices or roots)

D – Division,

M – Multiplication

S – Subtraction.

The BODMAS rule means that mathematical expressions with multiple operators need to be solved from left to right in this order. Division and Multiplication are considered as interchangeable and depend on which comes first in the expression, as are Addition and Subtraction.

Similarly, we have another acronym PEMDAS. The order of operations in PEMDAS is –

P stands for Parentheses

E stands for Exponents

M stands for Multiplication

D stands for Division

A stands for Addition

S stands for Subtraction

If we compare BOMAS and PEMDAS, we can see that both have similar order or precedence. In PEMDAS, the brackets have been represented as Parenthesis and orders have been represented as Exponents. Both use similar orders of precedence for simplifying mathematical expressions using order of operations. Let us now understand the use of the order of operations through an example.

Let us understand the order of operations using an example.

Example

Simplify the following expression using the Order of Operations –

6 + 3 – 4 x 2

Solution

We have been given the expression 6 + 3 – 4 x 2

It is important to note that there are three operations involved. Based on the order of operations, multiplication takes precedence over addition and subtraction so we will multiply first. Next, subtract then add since the operation of subtraction comes before addition from left to right.

Therefore, we have,

6 + 3 – 4 x 2

= 6 + 3 – 8 ( Here we have solved 4 x 2 which has given us the result 8 )

= Next, we will add 6 and 3 to get 9. So, we have,

6 + 3 – 8 = 9 – 8  = 1

Hence, 6 + 3 – 4 x 2 = 1

It is important to note here that the first alphabet in BODMAS is B which stands for Brackets. Similarly, in PEMDAS, P stands for parenthesis. Now, we know that there are different kinds of brackets or parentheses that are used in mathematics. Can these brackets be used randomly or do they have a particular order of precedence? Let us find out.

#### Use of Brackets in Order of Operations

We have just learnt about the precedence of fundamental operations of addition, subtraction, multiplication and division. According to it the order in which the operations are to be performed is first division then multiplication after which addition and finally subtraction but sometimes in complex operations require a set of operations to be performed prior to the other. For example, if we want the addition to be performed before a division or multiplication then we need to use a bracket.

Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression. Bracket indicates that the operations within it ought to be performed before the operations outside it. For example, the expression 24 ÷ 3 x 4 would generally be solved as –

24 ÷ 3 x 4

= 8 x 4 = 32

However, If we wish to multiply 3 and 4 first and then divide 24 by the resulting number we write the expression as

24 ÷  ( 3 x 4 )

Now, according to the PEMDAS rule, we will solve the brackets first to get,

24 ÷  ( 3 x 4 )

= 24 ÷  12

= 2

In complex expressions sometimes it is necessary to have (within) in the same time of (one within the another can be confusing for different types of brackets are used most commonly used) are

Brackets Name

( ) Parenthesis of Common Brackets

{ } Braces or Curly Brackets

[ ] Brackets or Square Brackets

It is important to note here that, the left part of each bracket symbol indicates the start of the bracket and the right part indicates the end of the bracket. In writing mathematical expressions consisting of more than one bracket, parenthesis is used in the innermost part followed by braces and these two are covered by square brackets.

Let us now understand the rules that need to be followed while removing brackets in an algebraic expression

#### Removal of Brackets

In order to simplify expressions involving more than one bracket, we will use the steps as explained below –

1. See whether the given expression contains vinculum or not. If vinculum is present then perform operations on the red otherwise go to the next step. A vinculum is a horizontal line placed over an expression to show that everything below the line is one group, for example $\overline{2\:x\:3}$.
2. Now, look for the innermost bracket and perform operations within it.
3. Remove the innermost bracket by using the following steps –
• If a bracket is preceded by a plus sign, remove it by writing its terms as they are.
• If a bracket is preceded by a minus sign change the positive sign within it to a negative sign and vice-versa.
• If there is no sign between a number and a grouping symbol then it means multiplication.
• If there is a number before some brackets then we multiply the number inside the brackets with the number outside the brackets.
4. Look for the next innermost bracket and perform operations in it. Remove the second in a directed by using the rules given in the above steps Continue this process till all the brackets are removed.

Let us understand the above steps using an example.

Example

Simplify: 37 – [ 5 + { 28 – ( 19 – 7 ) } ]

Solution

We have been given the expression, 37 – [ 5 + { 28 – ( 19 – 7 ) } ]

We will use the order of operations to solve the given expression.

First, we will remove the innermost bracket to get,

37 – [ 5 + { 28 – 12 } ] …………………… [ Removing the innermost bracket ( ) ]

Next, we will remove the curly braces to get

37 – [ 5 + 16 ] …………………………. [ Removing the curly braces ]

Now, we will remove the square brackets to get,

37 – 21 ………………………… [ removing the square brackets ]

Last, we will find the difference between 37 and 21 to get 16.

Hence simplification of 37 – [ 5 + { 28 – 12 } ] will result in 16.

Let us understand the above rules using an example.

There are certain rules that define the order of operations irrespective of the fact whether BODMAS or PEMDAS is used. Let us learn about these rules.

## Rules of Order of Operations

The following rules should be followed while finding the value of any algebraic expressions using the rules for order of operations –

1. Always start by calculating all expressions within parentheses

The first thing is that we should look for removing any grouping symbols in the algebraic expression. This means that at the top of the list, remember to always simplify everything inside the grouping symbols. Examples of grouping symbols are parentheses (  ), brackets, and braces {  }. For nested grouping symbols, work it out from the inside and out.

1. Simplify all the exponents such as square roots, squares, cube, and cube roots

After having removed the parenthesis, we come to the next step of solving all exponential values in the algebraic expression. Exponential expressions which may include root values such as square roots, squares, cube, and cube roots etc. are calculated or evaluated first before performing any of the four fundamental arithmetic operations, namely: addition, subtraction, multiplication, and division.

1. Perform the multiplication and the division starting from left to right

Next, multiply and/or divide whichever comes first from left to right before performing addition and subtraction. This tells us that multiplication and division have a higher level of importance than addition and subtraction.

1. Finally, do the addition and subtraction similarly, starting from left to right.

## Solved Examples

Example 1 Simplify 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

Solution We have been given the expression

95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

We will use the order of operations to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

95 – [ 144 ÷ ( 12 x 12 ) – ( – 4 ) – { 3 – 7 } ]

Next, we will remove the innermost bracket to get,

95 – [ 144 ÷ 144 + 4  – { 3 – 7 } ]

Now, we will remove the curly braces to get

95 – [ 144 ÷ 144 + 4  + 4 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

95 – [ 1 + 4  + 4 ]

Next, we will remove the square brackets to get,

95 – 9

= 86

Hence, simplification of 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ] = 86

Example 2 Simplify ( 1 + 3 ) 2 – 2 ( 4 – 7 )

Solution We have been given the expression

( 1 + 3 ) 2 – 2 ( 4 – 7 )

We can see that there is a parenthesis in the expression as well as an exponent. So, going by the order of operations, we will first solve the parenthesis to get,

( 4 ) 2  + 6

Next, we will solve the exponent in the expression to get,

16 + 6

Last, we are left with the addition operation after which we will get the result as 2.

Hence, simplification of ( 1 + 3 ) 2 – 2 ( 4 – 7 ) = 22

Example 3 Simplify 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Solution We have been given the expression

15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression; hence it needs to be solved first.

Solving the vinculum, we will get,

15 – ( – 5) { 4 – 4 } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Next, we will remove the innermost bracket to get,

15 + 5 x 0  ÷ [ 3 { 5 + 18 } ]

Now, we will remove the curly braces to get

15 + 0  ÷ [ 3  x 23 ]

Next, we will remove the square brackets to get,

15 + 0  ÷ 69

Now, it is important to see that we have two operations to be performed, division and addition. So, going by the order of operations, we will first perform division to get,

15 + 0

= 15

Hence, simplification of 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ] = 15

## Key Facts and Summary

1. Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations.
2. BODMAS is a sequence to perform operations in an arithmetic expression. It is an acronym where each alphabet represents a particular operation. BOMAS stands for – B – Brackets, O – Orders (powers/indices or roots), D – Division, M – Multiplication, A – Addition and S – Subtraction.
3. The PEMDAS Rule is a set of rules that prioritize the order of calculations, that is, which operation to perform first.
4. PEMDAS is an acronym where P stands for Parentheses, E stands for Exponents, M stands for Multiplication, D stands for Division, A stands for Addition and S stands for Subtraction.
5.  The rules of order of operations are –
• Always start by calculating all expressions within parentheses
• Simplify all the exponents such as square roots, squares, cube, and cube roots.
• Perform the multiplication and the division starting from left to right
• Finally, do the addition and subtraction similarly, starting from left to right.
1. Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
2. In writing mathematical expressions consisting of more than one bracket, parenthesis is used in the innermost part followed by braces and these two are covered by square brackets.