Order of Operations (BIDMAS/BODMAS)

• Mathematical Ambiguity: Without a defined order, an expression like $2 + 3 \times 4$ could result in $20$ (if adding first) or $14$ (if multiplying first). The order of operations removes this ambiguity.

• Operator Precedence: Higher-level operations like powers represent repeated multiplication, and multiplication represents repeated addition; therefore, they naturally carry more 'weight' and are processed earlier in the hierarchy.

• Left-to-Right Convention: When operations of equal priority (like Division and Multiplication) appear together, they are processed in the order they appear from left to right.

Step-by-Step Evaluation

• Step 1: Brackets: Solve everything inside parentheses first. If there are nested brackets, work from the innermost set outward.
• Step 2: Indices: Calculate all powers and roots.
• Step 3: Division and Multiplication: Perform these as they appear from left to right. Do NOT assume division always comes before multiplication.
• Step 4: Addition and Subtraction: Perform these as they appear from left to right.

Example Logic: In the expression $10 - 2^2 \times (1 + 1)$, you first solve the bracket $(2)$, then the index ($2^2 = 4$), then the multiplication ($4 \times 2 = 8$), and finally the subtraction ($10 - 8 = 2$).

• Equal Priority Groups: It is a common misconception that Division must happen before Multiplication because 'D' comes before 'M' in the acronym. They are actually equal in rank.

• Invisible Grouping: Fraction bars and square root symbols act as 'invisible brackets'. Everything above/below a fraction line or under a root must be treated as if it were in brackets.

• Show Intermediate Steps: Examiners look for the 'unrolling' of the expression. Write down the result of each priority level on a new line to avoid mental arithmetic errors.

• Calculator Entry: When using a scientific calculator, ensure you use the bracket keys for complex numerators or denominators to maintain the correct order.

• The Negative Square Trap: Always use brackets when squaring negative numbers on a calculator. Entering $-3^2$ may yield $-9$ (treating it as $-(3^2)$), whereas $(-3)^2$ correctly yields $9$.

• Sanity Check: After calculating, look at the operations. If you added before multiplying, your answer is likely incorrect unless the addition was inside brackets.

• The Left-to-Right Error: Students often fail to follow the left-to-right rule for subtraction and division. For example, $10 - 5 + 2$ is $7$, not $3$.

• Indices vs. Coefficients: Confusing $3x^2$ (where only $x$ is squared) with $(3x)^2$ (where both are squared).

• Distributive Property Overlap: Forgetting that a number outside a bracket implies multiplication, which must wait until the bracket's interior is resolved or the number is distributed.