Order of Operations (BIDMAS/BODMAS)

The Step-by-Step Evaluation

• Step 1: Brackets: Resolve everything inside parentheses first. If there are nested brackets, work from the innermost set outward.
• Step 2: Indices/Orders: Evaluate all powers ($x^2$), roots ($\sqrt{x}$), and reciprocals.
• Step 3: Division and Multiplication: Perform these from left to right as they appear in the expression. Do not assume division always comes before multiplication.
• Step 4: Addition and Subtraction: Perform these from left to right as the final step.

Handling 'Invisible' Brackets

• Fraction Bars: A fraction line acts as a grouping symbol. The entire numerator and entire denominator should be treated as if they are in brackets: $\frac{a+b}{c-d} = (a+b) \div (c-d)$.
• Radical Symbols: The horizontal bar of a square root symbol (the vinculum) groups the terms underneath it: $\sqrt{x+y} = \sqrt{(x+y)}$.

• Indices vs. Multiplication: An index represents repeated multiplication ($3^2 = 3 \times 3$), which is why it must be calculated before standard multiplication.

• Brackets vs. Operations: Brackets are not an operation themselves but a signal to change the default order of operations.

• The Underline Method: When solving long expressions, underline the specific operation you are about to perform and rewrite the rest of the expression exactly as it is to avoid losing terms.

• Left-to-Right Check: Always scan for strings of Addition/Subtraction or Multiplication/Division and ensure you process them in the order they are written, not the order of the letters in the acronym.

• Calculator Awareness: Modern scientific calculators follow BIDMAS automatically, but basic calculators may perform operations as they are entered. Always verify manual steps against a scientific calculator if permitted.

• The 'D before M' Trap: Many students believe they must divide before they multiply because 'D' comes before 'M' in the acronym. In reality, $12 \div 2 \times 3$ is $18$, not $2$.

• Subtraction Errors: Similar to division, subtraction is often incorrectly prioritized. In $10 - 5 + 2$, the correct answer is $7$ (left to right), but students often calculate $10 - 7 = 3$.

• Squaring Negatives: Without brackets, $-3^2$ means $-(3 \times 3) = -9$. With brackets, $(-3)^2$ means $(-3) \times (-3) = 9$. This is a frequent source of lost marks.