What Is the Order of Operations

What Is the Order of Operations?

The order of operations tells us what order to perform operations in.

A calculation may have several operations, such as: adding, subtracting, multiplying, dividing and squaring.

Why Do We Need the Order of Operations?

Imagine we wanted to find the answer to the calculation below:

This calculation contains two operations: adding and multiplying.

There are two orders to doing this calculation and two answers. Do we add then multiply, or multiply then add?

Order 1

Add the first two numbers, then multiply the result with the third number.

1 + 2 × 3 = 3 × 3 = 9

Order 2

Multiply the last two numbers, then add the result to the first number.

1 + 2 × 3 = 1 + 6 = 7

Which answer is the correct one?

It turns out the second order of operations is the correct one.

Luckily, there is a simple way to use the correct order.

BODMAS

BODMAS is an acronym for the order of operations. It stands for:

The order of operations is read from top to bottom. The operations with a curly bracket ({) are on the same level, and can be performed in any order.

  • Brackets. Evaluate brackets first.

  • Order. Evaluate exponents (such as squares and square roots) second.

  • Division and Multiplication. Evaluate numbers that are divided and multiplied third.

  • Addition and Subtraction. Evaluate numbers that are added and subtracted fourth.

How to Use the Order of Operations

Using the order of operations is easy.

Question

Find 2 + 32 − (8 × 2) ÷ 2.

Step-by-Step:

1

Brackets.

Evaluate expressions within brackets first.

In our example, there is one pair of brackets: (8 × 2) = 16.

2 + 32(8 × 2) ÷ 2

= 2 + 3216 ÷ 2

2

Order.

Evaluate numbers with exponents second.

In our example, there is one exponent: 32 = 9.

2 + 32 − 16 ÷ 2

= 2 + 9 − 16 ÷ 2

3

Division and Multiplication.

Evaluate numbers that are divided and multiplied third.

In our example, there is one division: 16 ÷ 2 = 8.

2 + 9 − 16 ÷ 2

= 2 + 9 − 8

4

Addition and Subtraction.

Evaluate numbers that are added and subtracted fourth.

In our example, there is one +'s and one . Addition and subtraction take the same precedence, so it does not matter which order we do them in. We will do them left to right.

2 + 9 − 8 = 11 − 8 \(\:\:\:\:\:\:\:\:\:\:\:\:\) as 2 + 9 = 11
11 − 8 = 3

Answer:

2 + 32 − (8 × 2) ÷ 2 = 3

Slider

The slider below shows another real example of how to use the order of operations.

Open the slider in a new tab

In another example, there is a pair of brackets which contains a long expression that itself needs to use the order of operations.

Open the slider in a new tab

See Also

How to add How to subtract How to multiply How to divide What is an exponent? What is an square number? What is a square root?