How To Simplify Expressions Using Order of Operations?

There are four basic arithmetic operations in arithmetic, plus (+), minus(-), multiply(×) and divide(÷). The expressions in mathematics can include some or all of these operations, for example, 30 - 18 ÷ 6 × 4 + 2. This expression involves +, -, × and ÷ together.

Such expressions with mixed operations of +, -, ×, ÷ and brackets (parenthesis) must be worked out in a specific order to get the result. There are some methods which sets order of operations known by different acronyms like BODMAS, BEDMAS, BIDMAS and PEMDAS. Each letter in BODMAS, PEMDAS, BEDMAS and BIDMAS represents an operation's name.

BODMAS is used in the UK, India and Australia.
PEMDAS in used USA.
BEDMAS is used in Canada and New Zealand
BIDMAS is used in the UK.

BODMAS

In BODMAS, operations are processed in the sequence of brackets, orders, division, multiplication, addition and subtraction.

Table of order of BODMAS operations
Order	BODMAS		Used for
1	B	Bracket	( ), { }, [ ]
2	O	Orders	$\sqrt{}$ , 3² (e.g.)
3	D	Division	÷
4	M	Multiplication	×
5	A	Addition	+
6	S	Subtraction	-

Brackets

The precedence of operation is given to brackets. The expressions inside brackets are always solved before any operations which are outside the brackets like addition, subtraction, division, multiplication or powers or roots. If there are multiple brackets surrounding each other then start with the innermost bracket and end with the outermost bracket.

Small bracket or round bracket ( )
Curly bracket { }
Large or square bracket [ ]

These three brackets must be solved in the following order:

Table of order of brackets
Order of brackets	Bracket name	Bracket symbol
1	Small or round bracket	( )
2	Curly bracket	{ }
3	Square or large bracket	[ ]

Example of bracket operation in BODMAS

Simplify [{(18 × 4) + (9 - 6)} - (6 × 3)] = ?

Step 1: Solve ( ) bracket

= [{72 + 3} - 18]

Step 2: Solve { } bracket

= [75 - 18]

Step 3: Solve [ ] bracket

= 57

How the brackets ( ), { } and [ ] are written together?

[ ] bracket must be the outermost bracket.
( ) bracket can only be the innermost bracket.
{ } bracket can surround ( ) bracket but not [ ] bracket.
[ ] can surround { } bracket. ( ) can not surround any of the { } and [ ] brackets.

Orders or powers or roots

After brackets, the second in precedence is Orders which includes operations of powers or roots. They are solved after brackets but before addition, subtraction, division and multiplication.

Example of power or root in BODMAS

(4³ - 10) + 9²

As, there is a term (4³ - 10) with brackets and 9² without brackets.

∴ the bracket term will be solved first.

Inside (4³ - 10), there is again a power and the minus operator.

So, the term with power, 4³, will be solved first than minus

∴ (4³ - 10) + 9² = (64 - 10) + 9²

Still need to solve bracket further with subtraction

= 54 + 9²

Now, there are two operators + and power. So, again solve first power term 9² before the addition

= 54 + 81

Perform the addition

= 135

$(\sqrt{121} - 6) - (2^{2} + 18)$

There are two terms with brackets $(\sqrt{121} - 6)$ and $(2^{2} + 18)$

∴ both bracket terms will be solved at the first step separately than the minus in between them. Each of them will also be checked for their own precedence order inside their brackets.

Roots in $(\sqrt{121} - 6)$ will be solved first than subtraction

Power in $(2^{2} + 18)$ will be solved first than addition

= (11 - 6) - (4 + 18)

Solve terms inside the brackets further

= 5 - 22

= - 17

Division

Third in order of operation is division operation. Division is performed after solving brackets and orders but before addition, subtraction and multiplication.

Example of division in BODMAS

49 ÷ 7 + (12² ÷ 4 - 11) - 5²

Division is at two places, inside and outside the bracket

But as the expression has bracket, so bracket will get the preference than division outside as per BODMAS.

Also, inside the bracket, the term with power will be solved first than division and subtraction.

= 49 ÷ 7 + (144 ÷ 4 - 11) - 5²

Still keep on solving the terms inside the bracket, by opting first division than subtraction

= 49 ÷ 7 + (36 - 11) - 5²

= 49 ÷ 7 + 25 - 5²

Now there exists four operations of ÷, +, - and power. Power will be solved first than division, addition and subtraction.

= 49 ÷ 7 + 25 - 25

Division will be solved before addition and subtraction.

= 7 + 25 - 25

Solve addition before subtraction because in BODMAS, A (addition) comes before S (subtraction).

= 32 - 25

= 7

Multiplication

In BODMAS, at fourth place is M, which is a multiplication operation. It comes after brackets, orders and division but before addition and subtraction.

Example of multiplication in BODMAS

15 × 400 ÷ (6³ - 16) + 8 × 4

The operation multiplication is at two places outside the bracket

But the brackets will be solved before solving the terms which are outside the bracket.

= 15 × 400 ÷ (216 - 16) + 8 × 4

= 15 × 400 ÷ 200 + 8 × 4

Now there are three operations of ÷, × and +.

As per BODMAS, division will get preference than × and +.

= 15 × 2 + 8 × 4

Solve × before + operator

= 30 + 32

Solve +

= 62

Addition

Fifth in BODMAS is A i.e. addition. It comes after brackets, orders, division and multiplication but before subtraction.

Example of addition in BODMAS

3 × 4³ - 8 + 4

There are four operators in the expression ×, power, - and +.

According to BODMAS, the four operations will be performed in the order of power, ×, + and -.

Solve powers

= 3 × 64 - 8 + 4

Solve ×

= 192 - 8 + 4

Solve +

= 196 - 8

Solve -

= 188

Subtraction

Sixth in BODMAS is subtraction operation. It is performed after the all operations brackets, orders, division, multiplication and addition.

Example of subtraction in BODMAS

75 - 17 × 2 + 16 ÷ 4 - 15

As per BODMAS, the order of operations for the expression will be ÷, × and -

So, first solve ÷

= 75 - 17 × 2 + 4 - 15

Solve ×

= 75 - 34 + 4 - 15

Solve +

= 79 - 34 - 15

Solve - from left to right

= 45 - 15

= 30

PEMDAS

PEMDAS is majorly used in USA. It processes operations in the order of:

Parenthesis
Exponents
Multiplication and division are equal in rank
Addition and subtraction are equal in rank

Table of order of PEMDAS operations
Order	PEMDAS		Used for
1	P	Parenthesis	( ), { }, [ ]
2	E	Exponents	$\sqrt{}$ , 5² (e.g.)
3	M	Multiplication	×
4	D	Division	÷
5	A	Addition	+
6	S	Subtraction	-

Multiplication and division are solved from left to right in an expression, whichever will come first, will be solved first also. Same is followed by addition and subtraction in PEMDAS.

Example of PEMDAS

√256 - 2² ÷ 4 × 10 ÷ [{(2 × 5) + (10 - 3)} - (3 × 5)] + 15 = ?

This expression has all operators like bracket, roots, exponents, multiplication, division, addition and subtraction.

First the terms with parenthesis in [{(2 × 5) + (10 - 3)} - (3 × 5)] will be solved.

Because there are multiple brackets inside the square bracket, so the they will be solved again in specific order of brackets, first ( ), second { } and third [ ]. Solve brackets from inside to outside.

Solve ( )

= √256 - 2² ÷ 4 × 10 ÷ [{(10) + (7)} - (15)] + 15

Solve { }

= √256 - 2² ÷ 4 × 10 ÷ [17 - 15] + 15

Solve [ ]

= √256 - 2² ÷ 4 × 10 ÷ 2 + 15

Now solve E for exponents and roots

= 16 - 4 ÷ 4 × 10 ÷ 2 + 15

Next solve M and D. Both M and D have equal ranks. Check from left to right, whichever comes first will be solved first also

Solve ÷

= 16 - 1 × 10 ÷ 2 + 15

Solve ×

= 16 - 10 ÷ 2 + 15

Solve ÷

= 16 - 5 + 15

Next solve A and S. Both A and S have equal ranks. Check from left to right, whichever comes first will be solved first also

Solve -

= 11 + 15

Solve +

= 26

The mnemonic for the PEMDAS is "Please Excuse My Dear Aunt Sally".

BEDMAS

BEDMAS is used in Canada and New Zealand. It differs from BODMAS by E only. E stands for exponents which is used to solve powers or roots.

Table of order of BEDMAS operations
Order	BEDMAS		Used for
1	B	Bracket	Used for	( ), { }, [ ]
2	E	Exponents	$\sqrt{}$ , 6² (e.g.)
3	D	Division	÷
4	M	Multiplication	×
5	A	Addition	+
6	S	Subtraction	-

BEDMAS acronyms defines the same order of operations like BODMAS. But BEDMAS moves left to right for MS (multiplication and division) whichever come first, will be solved first also. Both M and S are given equal precedence.

The same process applies for AS (addition and subtraction). A and S also get equal precedence.

BIDMAS

BIDMAS is mostly used in the UK. It uses I for indices which is used to solve powers and roots.

Table of order of BIDMAS operations
Order	BIDMAS		Used for
1	B	Bracket	Used for	( ), { }, [ ]
2	I	Indices	$\sqrt{}$ , 6² (e.g.)
3	D	Division	÷
4	M	Multiplication	×
5	A	Addition	+
6	S	Subtraction	-

BIDMAS acronyms define the same order of operations like BEDMAS. It also moves left to right for multiplication and division. M and S are given equal precedence. A and S too get equal precedence.

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