Order of Operations

Order of Operations — BIDMAS

When a mathematical expression contains more than one operation, the order in which those operations are performed can dramatically change the result. To ensure every mathematician — everywhere — gets the same answer to the same expression, an agreed convention called BIDMAS (or equivalently BODMAS) exists. Following this hierarchy is not optional: it is the universal rule for evaluating mathematical expressions.

Consider \( 3 + 4 \times 2 \). If you simply work left to right you get \( 7 \times 2 = 14 \). But the correct answer is \( 3 + 8 = 11 \), because multiplication must be performed before addition. BIDMAS tells you precisely which operation to tackle first.

The BIDMAS Hierarchy

Brackets, Indices (powers and roots), Division, Multiplication, Addition, Subtraction. Higher in the hierarchy means evaluated first.

Two Important Clarifications

Division and multiplication have equal priority. When both appear in the same expression (with no brackets distinguishing them), work from left to right. For example:

\[ 12 \div 4 \times 3 = 3 \times 3 = 9 quad \text{(not } 12 \div 12 = 1 \text{)} \]

Addition and subtraction have equal priority. Again, work left to right when both appear together:

\[ 10 - 3 + 2 = 7 + 2 = 9 quad \text{(not } 10 - 5 = 5 \text{)} \]

The square root sign \( \sqrt{\phantom{x}} \) and absolute value bars \( |\phantom{x}| \) act as grouping symbols, just like brackets — everything underneath the root or inside the bars must be evaluated before the root or absolute value is applied.

Worked Examples

No Brackets: \( 3 + 4 \times 2 - 1 \)

Step 1 — Multiplication first: \( 4 \times 2 = 8 \)

Step 2 — Addition and subtraction, left to right: \( 3 + 8 - 1 = 11 - 1 = 10 \)

\[ 3 + 4 \times 2 - 1 = 10 \]

Common error: Working left to right regardless: \( 3 + 4 = 7, 7 \times 2 = 14, 14 - 1 = 13 \). This is incorrect because multiplication must come before addition.

With Brackets: \( (3 + 4) \times (2 - 1)^2 \)

Step 1 — Brackets (innermost first): \( (3 + 4) = 7 \) and \( (2 - 1) = 1 \)

Step 2 — Indices: \( 1^2 = 1 \)

Step 3 — Multiplication: \( 7 \times 1 = 7 \)

\[ (3 + 4) \times (2 - 1)^2 = 7 \]

Notice how the brackets completely change the result compared to the unbracketed version. Brackets are the most powerful tool for controlling order of operations.

Powers & Roots: \( \sqrt{9 + 16} + 3^2 \times 2 \)

The expression under the square root acts as a bracket — evaluate it first.

Step 1 — Under the root (grouping): \( 9 + 16 = 25 \)

Step 2 — Indices and roots: \( \sqrt{25} = 5 \) and \( 3^2 = 9 \)

Step 3 — Multiplication: \( 9 \times 2 = 18 \)

Step 4 — Addition: \( 5 + 18 = 23 \)

\[ \sqrt{9 + 16} + 3^2 \times 2 = 23 \]

Complex Expression: \( 5 + (3 - 1)^2 \div 4 \times 3 \)

Step 1 — Brackets: \( (3 - 1) = 2 \)

Step 2 — Indices: \( 2^2 = 4 \)

Step 3 — Division and multiplication, left to right: \( 4 \div 4 = 1 \), then \( 1 \times 3 = 3 \)

Step 4 — Addition: \( 5 + 3 = 8 \)

\[ 5 + (3 - 1)^2 \div 4 \times 3 = 8 \]

Key point: Division and multiplication are evaluated left to right together in Step 3. Do not do all divisions before all multiplications — work from left to right through them as a group.

Key Takeaways

BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction) defines the universal order for evaluating expressions.
Brackets are evaluated first and are the most powerful tool for controlling the order — use them to override the default hierarchy.
Division and multiplication have equal priority and are evaluated left to right; the same applies to addition and subtraction.
The square root sign acts as a grouping symbol — everything under it must be evaluated before the root is applied.
A common error is working strictly left to right without applying the hierarchy — always check for higher-priority operations before proceeding.