Order of Operations — Grade 8 Maths (CAPS)

Do calculations in the wrong order and you can get the wrong answer, even with perfect arithmetic. BODMAS is the rule that fixes that.

Also known as: BIDMAS · BEDMAS · PEMDAS · order of operations

When a calculation has several operations, the order you do them in changes the answer. To make sure everyone gets the same result, maths follows a fixed order, remembered as BODMAS.

1What BODMAS stands for

The BODMAS ladder. Work down it in order; division/multiplication and addition/subtraction each rank equally.

B — Brackets first
O — Orders (powers and roots)
DM — Division and Multiplication, left to right
AS — Addition and Subtraction, left to right

The key subtlety: division and multiplication rank equally (do them left to right), and so do addition and subtraction. BODMAS is not strictly top-to-bottom for those pairs.

Worked ExampleWorked example: 3 + 4 × 2² − (5 − 1)

1
Brackets: (5 − 1) = 4, giving 3 + 4 × 2² − 4.
Always clear brackets first.
2
Orders: 2² = 4, giving 3 + 4 × 4 − 4.
Powers come before ×, ÷, + and −.
3
Multiply: 4 × 4 = 16, giving 3 + 16 − 4.
× before + and −.
4
Add and subtract left to right: 3 + 16 = 19, then 19 − 4 = 15.
The final answer is 15.

⚠️

The classic error: doing 3 + 4 first in '3 + 4 × 2'. Multiplication comes before addition, so it is 3 + 8 = 11, not 7 × 2 = 14.

💡

When in doubt, add your own brackets to show the order you intend — it keeps long calculations organised and easy to check.

2Division and multiplication: strictly left to right

Worked ExampleWorked example: 24 ÷ 4 × 2

1
÷ and × rank equally, so work from left to right.
BODMAS does not mean division before multiplication.
2
24 ÷ 4 = 6, giving 6 × 2.
3
6 × 2 = 12.
Doing 4 × 2 first gives 24 ÷ 8 = 3, which is wrong.

Worked ExampleWorked example: 20 − 8 + 5

1
− and + rank equally, so work from left to right.
2
20 − 8 = 12, then 12 + 5 = 17.
Adding first gives 20 − 13 = 7, which is wrong.

3Brackets inside brackets

When brackets are nested, work from the innermost pair outwards. Square brackets [ ] are simply a second kind of bracket, used to keep the levels apart.

Worked ExampleWorked example: 2 × [15 − (3 + 4)] + 6

1
Innermost brackets: 3 + 4 = 7, giving 2 × [15 − 7] + 6.
Start in the middle and work outwards.
2
Square brackets: 15 − 7 = 8, giving 2 × 8 + 6.
3
Multiply, then add: 16 + 6 = 22.

4Powers, roots and negative numbers

Worked ExampleWorked example: 5 + (−2)³ × 3

1
Orders: (−2)³ = (−2) × (−2) × (−2) = −8.
An odd power of a negative number is negative.
2
Multiply: −8 × 3 = −24, giving 5 + (−24).
3
5 − 24 = −19.

⚠️

(−3)² and −3² are not the same. (−3)² = (−3) × (−3) = 9, but in −3² only the 3 is squared: −3² = −(3 × 3) = −9.

Worked ExampleWorked example: √(9 + 16) compared with √9 + √16

1
A root sign groups everything under it, like brackets: √(9 + 16) = √25 = 5.
Work out what is under the root first.
2
√9 + √16 = 3 + 4 = 7.
Here each root is worked out separately.
3
The answers differ, so a root cannot be split over + or −.

5Exam favourite: insert brackets to make it true

Worked ExampleInsert one pair of brackets to make 12 − 4 ÷ 2 + 2 = 6 true

1
Without brackets: 12 − 2 + 2 = 12, not 6.
Division first, then left to right.
2
Try (12 − 4) ÷ 2 + 2 = 8 ÷ 2 + 2 = 4 + 2 = 6 ✓.
Brackets force the subtraction to happen first.

💡

A scientific calculator follows BODMAS by itself, but only if you type the brackets. For (18 + 6) ÷ (10 − 4) = 4, key in the brackets; typing 18 + 6 ÷ 10 − 4 gives 14,6.

BODMAS underpins every calculation you'll ever do, from Grade 8 arithmetic to Grade 12 calculus. Getting it automatic now prevents a whole category of avoidable mistakes.