Simplification & BODMAS

Simplification & BODMAS

Concept

BODMAS is the rule that resolves ambiguity in mathematical expressions. Without it, “3 + 4 × 2” could mean 14 (left to right) or 11 (multiply first). BODMAS is the agreed convention: Brackets first, then Orders (squares, cubes, roots), then Division and Multiplication (left to right, whichever comes first), then Addition and Subtraction (left to right).

Brackets come in three flavours that nest inside each other:

• Parentheses ( ) — the innermost
• Braces { } — the middle layer
• Square brackets [ ] — the outermost layer

When all three appear, solve from the inside out. For {2 + [3 × (4 + 5)]}: first the parentheses (4+5=9), then braces give {2 + [3 × 9]} = {2 + 27} = 29. Three layers, three steps.

Division and Multiplication are equal in priority — do whichever appears first as you read left to right. Same for Addition and Subtraction. A common mistake is treating DM as one block and AS as one block, then doing all M before all D. Wrong! 8 ÷ 4 × 2 = 4 (left to right), not 8 ÷ 8 = 1.

Key Formulas

Formula	Use
a ÷ b × c = (a ÷ b) × c	Left-to-right for DM
a − b + c = (a − b) + c	Left-to-right for AS
(a + b)² = a² + 2ab + b²	Special expansion
√(a × b) = √a × √b	Root distributivity
a × (b + c) = a×b + a×c	Distributive law

Worked Example

Q: Simplify: 48 ÷ {5 − [3 × (2 + 1)]}

Step 1: Innermost brackets → (2 + 1) = 3 Step 2: Now [3 × 3] = 9 Step 3: Now {5 − 9} = −4 Step 4: 48 ÷ (−4) = −12

Answer: −12

Common Errors

• Doing multiplication before division (or addition before subtraction) → remember they’re equal priority, go left to right
• Ignoring the negative sign when a bracket result is negative → −4 is perfectly valid as a divisor
• Forgetting to apply the exponent to a negative base: (−2)³ = −8, but −2³ = −8 anyway here — be careful with (−2)² = 4 vs −2² = −4