Order of Operations refers to the conventional ranking of mathematical operations used to evaluate expressions containing more than one operator. Without these rules, an expression like could be interpreted as either or , leading to global inconsistency.
The acronyms BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) and BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) serve as mnemonic devices for this hierarchy.
Brackets (or parentheses) are grouping symbols that isolate parts of an expression to be calculated first, effectively overriding the standard hierarchy.
Indices (also known as Orders, Powers, or Exponents) include operations like squaring (), cubing (), and finding roots (), which represent repeated multiplication or its inverse.
Step-by-Step Reduction: To evaluate an expression, identify the highest priority operation present, calculate its value, and rewrite the entire expression with that value substituted. Repeat this process until only a single number remains.
Handling Equal Priority: Division and Multiplication share the same level of priority. If an expression contains both, such as , you must work from left to right (), rather than assuming multiplication always comes first.
Addition and Subtraction Balance: Similarly, Addition and Subtraction are equal in rank. In the expression , you subtract first to get , then add to get .
Nested Grouping: If brackets are placed inside other brackets, always start with the innermost set and work your way outward.
Indices vs. Multiplication: Indices represent repeated multiplication (e.g., ), and because they are a higher-level operation, they must be resolved before any standard multiplication occurs in the expression.
The 'Invisible Brackets' Rule: Certain mathematical structures contain implied grouping that isn't explicitly written with parentheses. These must be treated as if they were inside brackets.
| Structure | Implied Meaning | Action Required |
|---|---|---|
| Fractions | Evaluate the entire numerator and entire denominator separately before dividing. | |
| Roots | Perform all operations under the root symbol before calculating the square root. | |
| Exponents | Calculate the value of the exponent expression before applying the power. |
Show Every Step: Examiners often award marks for the correct application of the order of operations even if a minor arithmetic error occurs. Writing out each intermediate line of the calculation is the best way to secure these marks.
The Calculator Trap: While modern scientific calculators follow BIDMAS automatically, older or basic models may calculate strictly from left to right. Always verify complex entries by adding manual brackets to ensure the calculator interprets the expression correctly.
Sanity Checking: After reaching a final answer, look back at the original numbers. If the result seems disproportionately large or small, re-check the 'Indices' and 'Multiplication' steps, as these are where the most significant errors usually occur.
Negative Number Caution: Be particularly careful with powers of negative numbers. For example, is , but is often interpreted as because the index has higher priority than the negative sign (which acts like a multiplication by ).
The 'M before D' Myth: A common mistake is believing that Multiplication must always happen before Division because 'M' comes before 'D' in the acronym. In reality, they are a single priority block and must be handled left-to-right.
Ignoring the Root Line: Students often calculate the root of the first number only (e.g., ) instead of following the top line of the root symbol to include all terms (e.g., ).
Premature Addition: In expressions like , students often add first. However, the is multiplying the bracketed term, so the multiplication must be performed before the addition.
