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Grouping Symbols and Order of Operations


In simplified mathematical statements that contain the same sort of operation, we focus on one operation at a time, moving from left to right. We can’t perform operations in the sequence they appear if an expression has more than one primary operation. Some operations must be completed first, followed by others. This means that each action has its own priority. Division, multiplication, addition, and subtraction are the operations that we conduct successively from left to right. When brackets are used in the expressions, however, we have a set of rules that determine the order in which the actions are performed. Let’s have a look at how this rule is established. Many distinct operations can be found in a mathematical problem. When a grouping symbol contains numbers, variables, or a math operation, it’s as if that component of the problem is saying, “Do me immediately!” Parentheses, braces, and braces are the most popular grouping symbols in mathematical problems. Those three serve that purpose in a math problem: to ensure that whatever is stored within those symbols receives the most focus.

To sets of numbers or variables, three main forms of grouping symbols are used: parentheses (), brackets [], and braces (letters). Parentheses are the most often used grouping symbols. Parentheses can be used before any actions.

  • We use these algebraic expressions is as follows
  • 3 + (3 – 5)
  • { 2 + 5( 3 * 5-2)-5}
  • [4^2 + 5 – 6 +(3-4)]

An algebraic question with several groups is organized using groupings symbols. Parentheses, brackets, braces, radicals, and fraction lines are algebraic grouping symbols that indicate where a group begins and finishes, as well as the order in which math procedures are applied. For anything outside the grouping symbol can act on words inside the grouping symbol, they must be operated on. None of the bracket types are stronger than the others or behave in a different way.


Mathematical brackets are indicators, such as braces, that are commonly used to form groups or indicate the order in which operations in an algebraic statement should be performed. However, several bracket characters have many applications in math and science. Brackets are always used in pairs for grouping. An opening bracket and a closing bracket will be used.

Brackets are used to clarify the sequence of operations, or the order in which various operations in a mathematical expression should be performed.

As an example, consider the phrase 4 + 4 * 5 – 1. There is only one correct answer to that expression, despite whatever we’ve wrote on Newsfeed. Arithmetic operations and divisions are performed from left to right before inclusions and subtractions, which are also performed from left to right. When we start with the multiplication, we’ll obtain 4+ 20 – 1 = 23.

5 – 5*5 + 6 

= 5 – 25 + 6

= 5 – 19 

=  – 17 answer 

What about if we wanted to start by adding and subtracting (and then multiplying the results)? Make use of brackets. The question now is: (4 + 4) * 5 – 1 = 8*5-1 = 39. The parentheses in this example instruct reader to perform actions in a different order than usual. They’re also sometimes utilized for visual clarity.

Type of Brackets

Brackets come in a variety of shapes and sizes.

{ } Curly brackets

[] Square brackets 

() Parenthesis or round brackets 

< > Angle Brackets

 (It’s worth noting that angle brackets resemble the “less than” and “greater than” indicators.)

If these three brackets used in the same question then first we give priority to parenthesis ()

Then we see the values of {} and solve them and at last we give attention to [] called square brackets.

What is the meaning of the term “order of operations”?

A regular approach that directs us about the computations to begin inside an argument containing several arithmetic operations is known as order of operation. It’s easy to make large blunders during computation if the order of operations isn’t consistent. Dependent on the inclusion of operators, brackets, multipliers, and other symbols for operations, there are a number of rules that define the sequence. BODMAS or PEMDAS is the name of this rule. Let’s look at the rule in greater depth.

Rules for order of operations

BODMAS and PEMDAS are two names for the terms BODMAS and PEMDAS.

BODMAS is a series of operations that can be used in an arithmetic computation. It’s an acronym in which each letter stands for a certain operation.

BOMAS stands for – Brackets – B 

Orders (powers/indices or roots) are letters that begin with the letter O.

M – Multiply, D – Division

A is for addition.

S stands for Subtraction.


The BODMAS rule states that mathematical expressions containing multiple operators must be solved in this order, from left to right. Division and Multiplication, like Addition and Subtraction, are interchangeable and rely on which occurs first in the phrase.

The word PEMDAS is commonly used in the United States, however it is known as BODMAS in India and the United Kingdom. The order of operations rules for braces, phases, addition, subtraction, multiply, and division is the same for both rules because there is no distinction amongst both. In different nations, the series of operations has been given different names, but the idea is the same.

Let’s use an example to better comprehend the order of operations

Example 1

Using the Order of Operations, simplify the following statement.

4 x 2 + 6 – 3 =


The formula 6 x 3 – 4 + 2 has been presented to us.

It’s worth noting that three operations are involved. Multiplication has precedence over addition and subtraction in terms of the order of operations, so we’ll start with that. Subtract first, then add from left to right, because subtraction comes before addition.

As a result,

18 – 4 + 2

= 18 – 2 

= 16

Example 2

4 + 3 x  6 – 10

= 4 + 18 – 10 

= 22 – 10 

= 12

Example 3

= 1 x 2 -10 +(19 -2) 0f 2        first we evaluate the bracket 

= 1 x 2 – 10 + 17 0f 2

= 1 x 2 – 10 + 34

 = 2 – 10 + 34 

= 2 – 24 

= 22 answer 

It’s worth noting that the first letter of BODMAS is B, which refers for Braces. “ p ” for parenthesis in PEMDAS as well. We now understand that there are various types of brackets or parenthesis used in mathematics. Is it possible to employ these brackets at random, or do they really have a specific sequence of priorities? Let’s see what happens.

It’s worth noting that the first letter of BODMAS is B, which refers for Braces. “ p ” for parenthesis in PEMDAS as well. We now understand that there are various types of brackets or parenthesis used in mathematics. Is it possible to employ these brackets at random, or do they really have a specific sequence of priorities? Let’s see what happens.

Brackets are used in the order of operations.

We just learned about the order in which the basic operations of addition, subtraction, multiplying, and divisions are performed. Accordingly, the operations should be conducted in the following order: division, multiplying, addition, and lastly subtraction, however complex processes may need a set of operations to be conducted before the others. As example, we need to include a bracket if we want the addition to happen before the division or multiplication. In complex statements, it is occasionally required to have (inside) at the same time of (two within the other can be misleading because multiple types of braces are regularly used).

Precedence of brackets

As, we already know we have three types of brackets named () Common Brackets in Parenthesis, Curly Brackets or Braces, Square Brackets or [] Brackets. It’s worth noting that the left half of each bracket sign represents the beginning of the bracket and the right half represents the end of the bracket. Parenthesis is employed in the innermost section of mathematical equations with more than one bracket, followed by braces, and these two are covered by brackets.

Brackets are also being removed.

We’ll utilize the procedures outlined below to simplify expressions involving more than one bracket

Check to see if the provided expression includes vinculum (A straight line drawn across an expression to indicate that everything below it is part of the same group). If vinculum is present, work on the red; otherwise, move on to the next phase. A vinculum is a horizontal line drawn over an expression to indicate that everything below it is part of the same group.

  • Now find the innermost bracket and conduct our procedures there.
  • Using the procedures below, remove the innermost bracket. –
  • If a plus sign before a bracket, delete it by putting the terms as they are.
  • If a minus sign before a bracket, the positive sign within it should be changed to a negative sign, and vice versa.
  • Multiplication is indicated when there is no sign between a number and a grouping symbol.
  • If a number appears before some braces, the number inside the brackets is multiplied by the number outside the brackets.
  • Locate the next closest bracket and carry out wer procedures there. Using the criteria outlined in the preceding phases, remove the second in a direction. Carry on in this manner until all of the brackets have been eliminated.


[2 + {4 – (4-6 x 2) + 6} -6]

= [ 2 + {4 – (4 – 12) + 6} – 6]

= [2 + { 4 – (-8) + 6} – 6]

= [ 2 + { 4 + 8 + 6} – 6]

= [2 +18 – 6]

= [ 20 – 6 ]

= 14 answer.

Instructions for order of operations

When utilizing the guidelines for order of operations to get the value of any algebraic expressions, the next rules should be followed:

  • Always start by calculating all expressions within parenthesis.
  • The first step is to check for any grouping symbols in the algebraic statement that can be removed. This implies that at the start of the list; make sure to keep anything inside the grouping symbols as simple as possible. Parentheses (), brackets (), and braces () are examples of grouping symbols. Work it out itself from the inside for nested grouping marks.
  • For example 

[2 + { 3 + 5 –{ 6 -4) – (4*2) – 5}+ 5]   is the example of grouping symbols in expression.

More examples are as follows

[5 –(5-7) + 6{6*7-7} – 8]

{3 – 6*1 /(5*3)}

  • Simplify all exponents, including square roots, squares, and cube roots.
  • After removing the parenthesis, the next step is to solve the algebraic expression for all exponential values. Before conducting any of the four fundamental arithmetic operations, such as addition, subtraction, multiplication, and division, proportional expressions, which would include root values such as square roots, squares, cubes, and cube roots, are calculated or evaluated first.
  • Reduce all exponents to their simplest form, includes square roots, squares, and cube roots.
  • After eliminating the parentheses, the mathematical expression must be solved for all exponentially values. Proportion expressions, which contain root values such as square roots, squares, cubes, and cube roots, are calculated or evaluated before any of the four fundamental arithmetic operations, such as addition, reduction, multiplication, and division.
  • 2 + { 2 – ( 8) -5 } 
  • = 2 + { 2 – 8 – 5}

          = 2 + ( -11)

            =  – 9


3 + [2 – 5 {4*6-(4+4)}]

= 3 + [2 – 5{ 4 *6 – 8}]

= 3 + [ 2 – 5{24 – 8}]

=3 + [ 2 – 5 {16}]

= 3 + [ 2 – 5*16]

= 3 + [ 2 – 80]

= 3 + [ – 78]

= 3 – 78 

= – 75


When there is no parenthesis, do we utilize the order of operations?

To simplify expressions, always use the order of operations. If no parenthesis are present, skip that step and continue on to the next. The same is true for any additional operations that are absent.

For example, to simplify the formula 6-2*4+2, use the order of operations.

There are no parentheses, so skip this step.

6- 9 + 2 is the number of exponents.

Multiplication/Division: Because there aren’t any, we  may skip this step.

6 – 7 (addition/subtraction)

 -1 required solution for given question. 

Can calculators have the ability to do order of operations?

No, usually calculators don’t comply with the requirement of operations, so be cautious when entering numbers. Be sure we stick to the order of events, even if it requires entering numbers in a random order than they appear on the screen. The solution for getting rid of this issue we can type brackets in calculators that will indicate the order of priority.

Solved Questions using parenthesis and grouping operator’s rule

Q: 1

3 + 4*5-[3-7(2-5)]

= 3 + 4 *5 – [ 3 – 7*(-3)]

= 3 + 4*5 – [ 3 + 21]

= 3 + 4*5 – 24

= 3 + 20 – 24

= 23 – 24 

= -1 which is the required answer.

Q: 2

10 + 4*4 – 3           here first we multiply because according to BODMAS rule we have to apply multiplication before addition and subtraction.

= 10 + 16 – 3 here first we do addition and then minus the resulting values.

= 26 – 3 

= 23 answer for the given example.

Q: 3

{3-4*4-(5-1)+2} – 5 

{ 3 – 4*4 – 4 + 2 }         here first we remove the parenthesis

{3 – 16 – 4 + 2}          here apply multiplication first according to BODMAS rule.

{ 3 -1 -2}

= { 2 – 2} 

= 0          hence this is our required answer.

Q: 4  

[2 – {4 – 1* 5 + (3 + 5)}] 

= [ 2 – { 4 – 1*5 + 8 )}] 

= [ 2 – { 4 – 5 + 8}] 

= [ 2 – { 4 + 3}]

=  [ 2 – 7]

= -5. Answer.

Q: 5  

58 x 20 + 32 

= 58 x 20 + 32 Execute the exponentiation operation = 1160 + 32. 

 Carry out the addition operation 1160 + 9 . now. Last but not least, conduct the addition operation= 1192. 

 As a result, the value of answer is 1192.


The order of events establishes a coherent computation sequence. We could get various answers to the identical calculation problem if we didn’t know the order of operations. When evaluating arithmetic expressions, the following order of operations must be followed: First, make all operations inside parenthesis as simple as possible. Then, working from left to right, execute all multiplications and divisions, Then, working from left to right, complete all additions and subtractions. If the question contains a fraction bar, all computations above and below the fraction bar must be completed before splitting the fraction even by denominator.

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