Algebra - Brackets
- Definition of Bracket
- Types of Brackets
- How to Solve an Expression with Brackets?
- Solved Examples
- Practice Questions
- Frequently Asked Questions
Definition of Bracket
Brackets are special symbols as shown: ( ), { }, and [ ]. Brackets are used in maths to combine certain numbers or expressions. Additionally, they are used if precedence has to be given to a certain expression over others. Following are some expressions using brackets.
Types of Brackets
Different types of brackets are used to distinguish among different groups of brackets. All the brackets are written in pairs which means that if there is an opening bracket then there must also be a closing bracket.
We are going to discuss three types of brackets here:
`1`. Round Brackets or Parentheses: The parentheses are written as (). These are the most commonly used brackets in maths. While evaluating expressions, parentheses are used for grouping and for specifying the precedence of expressions.
- Parentheses can also be used to write negative numbers. For example, negative `60` can be written as `-(60)`.
- It can be used to give precedence to some expressions/numbers over others. For example, in the expression `3×(2 + 5)`, `(2 + 5)` is solved first. So, the expression in turn becomes `3×(7)` which becomes `21`.
- When two numbers are written with parentheses, it means that numbers are to be multiplied. For example, `(42)(2)` means `42` multiplied by `2`. So, the answer is `84`.
- Parentheses are used to denote the input variable in a function, like the use of parentheses in `f(x)` where `x` serves as the input variable to function `f`.
- To represent an ordered pair of a point on a coordinate plane, we use parentheses to enclose the `x` and `y` values of the coordinate. For example: `(7, 4)`.
`2`. Curly Brackets: The pair of brackets written like { and } is called curly brackets. Curly brackets are also called braces.
- Curly brackets are used to denote a set. For example: `{a, b, c}`
- Curly brackets can also be used to group large expressions like `8[1+(3+{2+4})]`
`3`. Square Brackets: Square brackets are used when we want to precede a set of operations over others in an expression that already has parentheses. For example: `-12[90 - (9 + 7) + (6 + 2)]`.
How to Solve an Expression with Brackets?
An expression or equation involving multiple brackets can be solved using the order of operations of brackets. Following are the steps that are used to solve an expression with multiple brackets.
- First, expressions inside (Parentheses) are solved. After that, expressions inside {Curly brackets} are solved. Finally, expressions inside [Square brackets] are solved.
- After solving the brackets, in the next step, exponents are solved.
- In the third step, multiplication and division operations are performed. Out of division and multiplication whichever operator comes first (going from left to right) that particular operation is performed first.
- After solving the multiplication and division operations, addition and subtraction operations are performed whichever comes first (going from left to right).
The acronym PEMDAS can be used to remember the above order of operations. The table below shows the order going from left to right.
Solved Examples
Example `1`: Evaluate: `5× 4 ÷ 10`
Solution:
`5×4 ÷ 10`
`= 20 ÷ 10`
`= 2`
Example `2`: Evaluate the expression: `74 + [(9-1)× 5× 4]`
Solution:
Step `1`: Simplify the expression inside the parentheses first `(9-1)` which becomes `8`
`= 74 + [8× 5× 4]`
Step `2`: Simplify the expression inside the square brackets going from left to right
`= 74 + [40× 4]`
`= 74 + [160]`
Step `3`: Add the two numbers `74` and `160`
`= 234`
The answer is `234`.
Example `3`: Evaluate the expression `{(8-2)\times 4} ÷ 6`.
Solution:
Step `1`: Simplify the expression inside the parentheses first, `(8 - 2)` which becomes `6`
`= {6× 4} ÷ 6`
Step `2`: Next, simplify the expression inside the curly braces `{6× 4}` which becomes `24`
`= 24 ÷ 6`
Step `3`: Finally, divide the number `24` by `6`
`= 4`
The answer is `4`.
Example `4`: Evaluate: `[ 66 + { (90 - 45)× (8-2) } ]`
Solution:
Step `1`: Simplify the expressions inside the parentheses first : `(90 - 45)` and `(8× 2)` which gives `45` and `16` respectively.
`= [ 66 + { 45× 6 } ]`
Step `2`: Next, simplify the expression inside the curly brackets: `{ 45× 6 }`
`= [ 66 + 270 ]`
Step `3`: Finally, solve the square brackets `[ 66 + 270 ]`
`= 336`
The answer is `336`.
Example `5`: Evaluate: `8(90) +9(80 + 2× 4)`
Solution:
Step `1`: Simplify the expression inside the parentheses first : `(80 + 2× 4)`
`= ( 80 + 8 )`
Step `2`: Add `( 80 + 8 )`
`= 88`
Step `3`: Multiply `8` by `90` before the addition sign: `8(90)`
`= 8× 90`
`= 720`
Step `4`: Multiply `9` with the number obtained in step `2` i.e. multiply `9` with` 88`
`= 9× 88`
`=792`
Step `5`: Add the answers obtained in steps `3` and step `4` i.e. `720` and `792`
`= 1512`
The answer is `1512`.
Practice Problems
Q`1`. Evaluate the expression: `(81÷ 3 + { 9 + 90× [8 + (6-1)]})`
- `1208`
- `1206`
- `1204`
- `1202`
Answer: b
Q`2`. Evaluate the expression: `9(80) + 7(60) + 5(40)`
- `1320`
- `1330`
- `1340`
- `1350`
Answer: c
Q`3`. Evaluate the expression: `[ 6 + { 6 + [ 46 × ( 6 + 6 ) ] } ]`
- `558`
- `560`
- `562`
- `564`
Answer: d
Q`4`. Evaluate the expression: `243 ÷ 27 + ((16 - 2) + 216)`
- `239`
- `245`
- `212`
- `228`
Answer: a
Q`5`. Evaluate the expression: `12 + {3456 ÷ 6} - 10`
- `568`
- `570`
- `578`
- `612`
Answer: c
Frequently Asked Questions
Q`1`. What is the purpose of square brackets [ ]?
Answer: Square brackets are commonly used in mathematics to denote intervals, matrices, and arrays. In an expression, square brackets are used when we want to precede a set of operations over others in an expression that already has parentheses.
Q`2`. When should I use parentheses () in mathematical expressions?
Answer: Parentheses are used to indicate the order of operations in mathematical expressions. They clarify which operations should be performed first.
Q`3`. How do I determine the priority of different types of brackets in an expression?
Answer: In the order of operations, parentheses ( ) have the highest priority, followed by curly brackets { }, and then square brackets [ ]. Evaluate expressions within the innermost brackets first and proceed outward.