**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.

**Definition**

BODMAS 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, where

B stands for Brackets

O stands for Of

D stands for Division ( ÷ )

M stands for Multiplication ( x )

A stands for Addition ( + )

S stands for subtraction ( – )

**Understanding BODMAS**

Understanding BODAMS is integral to the understanding of the use of grouping symbols in expressions. This is because it is the BODMAS rule that defines the order in which the operations are to be performed in the case of mathematical expressions. In other words, it defines the order in which operations on numbers are to be performed when there are situations where we have two more than two operations in an expression.

Let us consider an example. Suppose we want to find the value of 3 + 5 x 2.

Here, we have two operators, namely ( x ) and ( + ). We know that in mathematics, we can perform operations between two numbers only in one go. If we have more than 2 numbers, the result from the first operation is used for performing the next operation. So, in the above example, there are two operations to be performed, ( 3 + 5 ) and ( 5 x 2 ). We need to decide, which one to start with. This is where the MDAS rule comes into play.

From this rule, we can see that Multiplication precedes addition. Therefore, we will perform the operation multiplication first. The result thus obtained will be added to 3. The steps involved in this process will be –

3 + 5 x 2

= 3 + 10

= 13

**Hence, 3 + 5 x 2 = 13**

**How to use BODMAS in arithmetic expressions**

It is important to note that when there are no special grouping symbols, math problems are solved from left to right. However, when you have grouping symbols involved we need to follow the order of operations as we have discussed above. 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.

**BODMAS is 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 order of operations, 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.

Some other important points to note are –

- Parentheses are used in math to show a part of a math expression or equation that must be solved first before any other calculations are done. The part between the two parentheses is treated like one number; the answer replaces the expression in the larger math equation.
- For complicated problems, brackets can be used to enclose sections of the problem that already include parentheses to further separate sections.
- For extremely complicated problems, braces can be used to enclose sections that already include brackets and parentheses.

**Removal of Brackets**

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

- 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 $\overline{2 x 3}$.
- Now, look for the innermost bracket and perform operations within it.
- 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.

- 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.**

**Using BODMAS Rules to Simplify Mathematical Expressions**

Now, that we have understood what we mean by grouping symbols and mathematical expressions let us learn about the steps involved in solving mathematical expressions involving BODMAS. The steps involved are –

- 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.

- 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.

- 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.

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

Let us understand it using an example.

**Example** 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 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,

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**

Let us consider another example.

**Example** Find the value of the expression 4 [ 10 + 15 ÷ 5 × 4 – 2 × 2 )

**Solution** The steps used to solve the given expression can be tabulated as

Brackets Of | 4 ( 10 + 15 ÷ 5 × 4 – 2 × 2 ) |

Division | 4 ( 10 + 15 ÷ 5 x 4 – 2 × 2 ) |

Multiplication | 4 ( 10 + 3 × 4 – 2 × 2 ) |

Addition | 4 ( 10 + 12 – 4 ) |

Subtraction | 4 ( 22 – 4 ) |

Answer | = 4 x 18 |

72 |

**Solved Examples**

**Example 1** Simplify: 197 – [1/9 { 42 + (56 – $\overline{8+9}$ ) } +108 ]

**Solution** We have been given the algebraic expression,

197 – [1/9 { 42 + (56 – $\overline{8+9}$ ) } +108 ]

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,

197 – [1/9 { 42 + (56 – 17 ) } +108 ]

Next, we will remove the innermost bracket to get,

197 – [1/9 { 42 + 39 } +108 ]

Now, we will remove the curly braces to get

197 – [81 / 9 +108 ]

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,

197 – [9 + 108 ]

Next, we will remove the square brackets to get,

197 – 117

Last, we just need to find the difference of the remaining two values to get,

80.

**Hence, simplification of 197 – [1/9 { 42 + (56 – **$\overline{8+9}$** ) } +108 ] = 80.**

**Example 2** 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 PEMDAS rule, 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**

- 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.
- BODMAS is the basic order of operations where, B stands for Brackets, O for Of, D for Division( ÷ ), M stands for Multiplication ( x ), A stands for Addition ( + ), S stands for subtraction ( – ).
- BODMAS are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
- The rules of using BODMAS to simplify arithmetic expressions 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.

- Brackets are used to provide clarity in the order of operations, the order in which several operations should be done in a mathematical expression.
- 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.

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