## What is Brackets?

You must have seen different symbols like these: (, ), [, ], {, and } in your math books. These symbols are called brackets. Brackets in mathematics serve a very important purpose; these symbols help us group different expressions or numbers together. Brackets imply that the thing or expression enclosed by them is to be given higher precedence over other things.

## Different Kinds of Brackets

Generally, three kinds of brackets are used in mathematics,

- Parentheses or Round Brackets, ( )
- Curly or Brace Brackets { }
- Square or Box Brackets [ ]

## Parentheses Brackets

These are also known as the round brackets and written as ( ). These are the most common types of brackets. They are used for grouping different values and equations together.

When used simply around numbers, the round brackets denote multiplication.

For example : (3)(4) = 12

They can also be used to write negative integers in mathematical expressions.

For example 5 + (−4) = 1

Parentheses can also be used to separate out numbers from their exponents. For example: {2}$^{-3}$

Examples: (2 + 4), 5(111), 25 – (12 + 8), etc.

## Curly Brackets

Like Parentheses, curly brackets are also used to group various mathematical components; however, curly brackets are also used to depict sets or to write nested expressions. Examples:

[4 + [3 $\times$ (- 2)] – [{(4 $\times$ 6) + (14 $\div$ 7)} – (- 3)],

[{12 − (12 − 2) } + (5 − 7)] + 9, etc.

## Square Brackets

Square brackets are generally used to distinguish between sub-expressions of a complex mathematical expression.

Examples: [100 – (3 – 1) + (7 x 8)], 10 x [(4 – 2) x ( 4 x 2)], etc.

## Order of Operations of Brackets

When we evaluate a mathematical expression that is made up of different brackets, we have to follow certain rules. This is called the rules of operation or order of operation of brackets.

- The general order of operation of the bracket can be illustrated as [ { ( ) } ]; this means that in a given problem, you would have to simplify the values in the innermost bracket first. This means that first ( ) brackets will be solved, following which, { } brackets are solved and finally [ ] brackets.
- The second step in solving these problems is to look for an exponent; if there is any, solve it first.
- In the third step, we look for expressions with multiplication or division operators. If both the operators are present, we check the expression from left to right. Whichever operator comes first, we solve that operator first.

For example, in the expression, 10 6 ÷ 5, we check from left to right, since multiplication comes first so we solve multiplication first and then division.

10 $\times$ 6 ÷ 5

= 60 ÷ 5

= 12

- In the fourth and last step, we look for numbers that need to be added or subtracted. We follow the same instruction if both the operators are present, we look from left to right in the expression, and whichever operator comes first, we solve that expression first. But if the operations are in brackets, we always solve the brackets first since brackets have the utmost precedence.

To remember the above mention steps, we can use the acronym PEMDAS,

P – Parentheses,

E – Exponents

M – Multiplication

D – Division

A – Addition

S – Subtraction.

let’s use pemdas to evaluate the expression

100 − [(3 – 1) + (7 x 8)]

Step 1: Solve the brackets. Follow the order of solving round brackets ( ) first, then curly brackets { }, and then square brackets [ ].

= 100 − [(2) + (56)]

= 100 − 58

Step 2: No exponent in the given expression.

Step 3: No multiplication or division in the given expression.

Step 4: Solve the subtraction.

= 100 − 58

= 42

## Solved Examples

**Question 1: Find the value of the expression: (5 + 4) − (3 − 2).**

**Answer:** The given expression is,

(5 + 4) **−** (3 **−** 2),

Step 1: Solving the values in the brackets,

(9) **−** (1),

Thus, the answer is (9) **−** (1) = 8.

**Question 2: ****Find the value of the expression: ****{(7 − 2) × 3} ÷ 5**

Answer: The given equation is,

{(7 − 2) × 3} ÷ 5

Step 1: Solving the parentheses

{(7 − 2) × 3} ÷ 5

= {5 × 3} ÷ 5

Solving the curly bracket

= {15} ÷ 5

= 15 ÷ 5

= 3

**Question 3: ****Find the value of the expression: ****(12 ÷ 6) × (4 − 2)**

Solution:

The given equation is,

(12 ÷ 6) × (4 − 2)

Solving the values in the brackets,

(2) x (2)

Thus, the answer is (2) x (2) = 4

**Question 4: Find the value of the expression: [120 + { (3 x 4) + (4 − 2) − 1 } + 20]**

**Answer:** Following the PEMDAS rule, first,

Step 1: We solve the values in ( ) brackets,

[120 + { (3 x 4) + (4 − 2) − 1 } + 20 ]

= [ 120 + { (12 ) + ( 2 ) **−** 1 } + 20 ],

Now we solve the values inside the { } brackets,

= [ 120 + { 13 } + 20 ],

Finally, add all the values in the [ ] bracket,

The answer is 153.

## Practice Problems

## Brackets

### 1Solve: [{(22 + 33) × 42} −(20 ÷ 5)]

Step 1: Solve all brackets keeping the precedence in mind.

[{(22 + 33) × 42} −(20 ÷ 5)]

= [{(4 + 27) × 16} −(4)]

= [{(31) × 16} −(4)]

= [{31 × 16} −4]

= [496 −4]

= 492

### 2What is the right representation of the order of operation in brackets?

[ { ( ) } ] is the correct representation of order of operation in brackets.

### 3$\Biggr [\bigg\{ \bigg( \frac{1}{2} \bigg)^2 \bigg\}^{-3} \Biggr]^2$

$\Biggr[\bigg\{ \bigg(\frac{1}{2}\bigg)^{2}\bigg\}^{-3}\Biggr]^{2} = \Biggr[\bigg\{\frac{1}{4}\bigg\}^{-3}\Biggr]^{2} = \Biggr[\bigg\{\frac{4}{1}\bigg\}^{3}\Biggr]^{2} = [64]^{2} = 4,096$

### 4Solve this expression, 12 + (5 + 3),

the correct answer is 20.

## Frequently Asked Questions

**Why are brackets important in mathematics?**

Brackets are very important parts of a mathematical equation; they separate different mathematical expressions from each other and help set the priority for expressions that need to be solved first.

**Is PEMDAS the only method to solve bracket problems?**

BODMAS is a different acronym for PEMDAS, where B stands for Bracket, O for Of or Exponents, D for Division, M for Multiplication, A for Addition, and S for Subtraction. Any expression is considered correctly solved if they have followed the PEMDAS or BODMAS rule.

**Are there any more kinds of brackets?**

Angle Brackets are also used in various mathematical expressions; they are represented with〈 〉. The angle brackets are used to represent a list of numbers or a sequence of numbers.

**What are some other applications of brackets?**

Brackets are also used to define the coordinates of a point on a map or to describe the variable of a function.