# What Is The Order of Operations?

An operation is to add, subtract, multiply, divide, square and so on. Sometimes, there are many different operations in a mathematical expression. For example:

Here there is a subtraction and a multiplication. There will be a different answer to the expression depending on what order the operations are done in. To decide what order, we use the order of operations. The order is given by BODMAS.

# What Is BODMAS?

BODMAS is an acronym, telling the order that operations should be done in. It stands for:

The order of operations is read from top to bottom - although the operations with a curly bracket '{' are on the same level.

Step 1
Brackets ( ). Simplify the expression within any brackets.

Step 2
Orders. Simplify powers, exponents and roots. For example, 22, 53, or √4.

Step 3
Division ÷ and/or Multiplication ×. Simplify the expression by performing divisions and multiplications.

Step 4
Addtion + and/or Subtraction -. Simplify the expression by performing addtions and subtractions.

# A Real Example of Using Order of Operations

Consider the expression at the top of the page. First, simplify the expression without using the order of operations. Take the operations in the order they are written

Without Using Order of Operations.
8 - 5 x 2       Subtraction       8 - 5 = 3
3 x 2            Multiplication
6                  Incorrect

Now, simplify the expression using the order of operations. Looking at BODMAS, M comes before S. Multiplcation before Subtraction.

Using Order of Operations.
8 - 5 x 2       Multiplication   5 × 2 = 10
8 - 10           Subtraction
-2                 Correct

Not using the order of operations resulted in an incorrect answer, even for the simple expression used above.

In the slider below, BODMAS is used to simplify a more complex expression. In the next slider, the case where there are several operations within the bracket is shown.
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# BODMAS

Another way of writing BODMAS is as below:

This makes it clear that there are 4 distinct steps or levels in the order of operations.

Brackets first.

Orders second.

Division and Multiplication are on the same level. It does not matter in which order divisions and/or multiplications take place. This is because of the commutative property of multiplication and division.

Addition and Subtraction are also on the same level, and can be done in any order. This is because of the commutative property of addition and subtraction.

# BRACKETS

Brackets are the first operation that is performed according to BODMAS. The expression within the brackets should be simplified first.

However, the expression within the brackets may itself contain brackets, orders, divisions, multiplications, additions and subtractions.

In the case, use BODMAS on the expression within the Bracket, simplifying the bracket expression, before applying ODMAS to the rest of the original expression.

BODMAS, BIDMAS, BEDMAS, PEMDAS

BODMAS can also be remembered as BIDMAS or BEDMAS - where instead of O for Order, there is I for Indices or E for Exponent. They are all different words for the same thing.

The acronym PEMDAS is also sometimes used: Parentheses, Exponent, Multiplication, Division, Addition, Subtraction.

Parentheses is another word for brackets.

A useful memory device for PEMDAS is "Please Excuse My Dear Aunt Sally."