Order of Operations (BIDMAS/BODMAS)

Order of operations is the agreed convention for evaluating expressions containing more than one operation. It ensures that a calculation has one consistent meaning by prioritizing grouping symbols, powers, multiplication and division, and then addition and subtraction, with ties handled from left to right. Understanding BIDMAS or BODMAS is essential for arithmetic, algebra, calculator use, and interpreting fractions, roots, and negative numbers correctly.

• Order of operations is the standard rule for deciding the sequence in which parts of a calculation are evaluated. It matters because expressions such as $3 + 4 \times 5$ would otherwise have more than one possible answer, so mathematics uses a shared convention to remove ambiguity.
• BIDMAS or BODMAS is a memory aid for the sequence: Brackets, Indices or Orders, Division and Multiplication, Addition and Subtraction. The acronym helps recall the priority levels, but the deeper idea is that some operations create tightly bound structures that must be completed before looser ones.
• Brackets such as $(\ )$ act as grouping symbols and tell you to treat part of an expression as a single unit first. This is why $2(3+4)$ means multiply by the complete result of the bracket, not just the first number inside it.
• Indices, powers, and roots belong to the same priority level because they describe repeated multiplication or its inverse. For example, $a^n$ means repeated multiplication, while $\sqrt{x}$ asks for a number whose square gives $x$, so both must be resolved before ordinary multiplication or addition.
• Division and multiplication share the same level of priority, and addition and subtraction also share the same level. When operations have equal priority, you work from left to right, because division is not automatically done before multiplication, and subtraction is not automatically done before addition.

• The purpose of BIDMAS/BODMAS is consistency, not convenience. If every person could choose a different order, algebraic expressions, formulas, and calculator entries would not have a single agreed meaning, so the rule acts as a shared language of calculation.
• Operations have different binding strength, meaning some parts of an expression naturally stick together more tightly than others. For instance, in $2 + 3 \times 4$, the multiplication forms the quantity $3 \times 4$ first, and only then is that result combined with $2$.
• Equal-priority operations are handled left to right to avoid inventing extra hierarchy where none exists. This means expressions like $a \div b \times c$ are interpreted as $(a \div b) \times c$, not $a \div (b \times c)$, unless brackets explicitly say otherwise.
• Fractions represent division and often contain invisible grouping. In a fraction such as $\frac{2+5}{7-2}$, the numerator and denominator each behave like bracketed expressions, so the fraction bar groups the top and bottom before the final division occurs.
• Root symbols also create grouping, because everything under the root sign belongs to one operation. For example, $\sqrt{9+16}$ means the square root of the entire quantity $9+16$, not just the square root of $9$ followed by adding $16$.

Step-by-step method

• Step 1: Identify grouping first by locating brackets, fraction bars, root bars, and any other notation that encloses a whole expression. This prevents you from breaking apart terms that are meant to be treated as one unit.
• Step 2: Evaluate powers, indices, and roots after grouped parts have been simplified. This works because expressions such as $4^2$ or $\sqrt{25}$ define single values before they can sensibly be multiplied, divided, added, or subtracted.
• Step 3: Work through multiplication and division from left to right without jumping ahead based on preference. A useful mental model is to scan the expression in order and simplify one operation at a time while rewriting the new line clearly.
• Step 4: Finish with addition and subtraction from left to right once all higher-priority operations are complete. At this stage the expression has been reduced to combined terms, so only the loosest level of calculation remains.
• Step 5: Rewrite after each stage rather than trying to hold too many changes in your head. This reduces arithmetic slips, sign errors, and mistaken cancellations, especially in longer expressions.

Working rule: Group first, powers next, then $\times$ and $\div$ left to right, and finally $+$ and $-$ left to right.

Calculator technique

• Modern calculators usually follow the written order of operations, so you can often type an expression as it appears. Even so, adding explicit brackets is wise when the structure is complicated, because it makes your intention unambiguous and easier to check.
• Older or simpler calculators may need extra brackets to reproduce the correct structure. This is especially important for fractions, negative numbers, and powers, where small input mistakes can change the entire meaning of the expression.

Operations with equal priority

• Multiplication and division are peers, not a fixed sequence where one always comes first. You choose between them based on which appears first from left to right, because they are inverse operations at the same structural level.
• Addition and subtraction are also peers, so they are handled left to right once the higher-priority work is complete. Treating subtraction as a separate later step often causes incorrect regrouping and sign mistakes.

Visible and invisible grouping

• Brackets are visible grouping symbols, but they are not the only way expressions are grouped. Fraction bars, root bars, and even function notation can create grouped units that must be respected before moving to later steps.
• A negative sign is not the same as subtraction in every context. In $(-3)^2$, the negative number is grouped as the base, while in $-3^2$ the power applies first to $3$, and the negative sign is applied afterward.

Key distinction: $(-a)^2$ and $-a^2$ are generally different expressions because the brackets change what the power applies to.

• Read the structure before calculating. A quick scan for brackets, powers, fraction bars, roots, and negative signs often prevents the most common exam errors, because many mistakes happen before any arithmetic begins.
• Show stages clearly by rewriting the expression line by line. Examiners can often see your method from the sequence of simplifications, and this also helps you check whether each step respects the correct priority.
• Use estimation or a reasonableness check at the end. If an expression contains mostly positive terms and your answer is unexpectedly negative, or if a small change in one term seems to create a huge result, recheck the order you used.
• Be especially careful with calculator entry, because even when the calculator knows BIDMAS/BODMAS, it only follows what you typed. If the structure is not obvious, extra brackets are a strategic safeguard rather than a sign of weakness.
• Memorize the left-to-right rule for equal levels, since exams often test this idea indirectly. Many incorrect answers come from treating multiplication as always before division or addition as always before subtraction, which is not how the convention works.

Exam habit: Before pressing equals, ask "What is grouped, what is powered, and which equal-priority operations must be read left to right?"

• A common misconception is that the letters in BIDMAS/BODMAS force a strict internal order within each pair, such as always multiplying before dividing. In fact, multiplication and division share one level, so the correct rule is to evaluate them from left to right.
• Students often ignore invisible brackets in fractions and roots. This leads to splitting the numerator, denominator, or radicand incorrectly, even though the notation itself shows that the whole grouped part belongs to one operation.
• Negative signs are frequently mishandled around powers. The expressions $(-a)^2$ and $-a^2$ look similar, but the first squares the negative number while the second applies the negative sign after squaring $a$.
• Another pitfall is mentally combining terms too early, such as adding before finishing multiplication or division. This usually happens when students focus on reading order rather than structural priority, so slowing down and marking each level is helpful.
• Calculator trust can become a trap if the input does not match the intended structure. Even a correct calculator will return a wrong result when brackets are omitted around negative values, numerators, denominators, or exponents.

• Order of operations is foundational for algebra, because algebraic expressions follow the same structural rules as numerical ones. Simplifying $3x + 2x^2$ or substituting values into formulas requires the same attention to grouping, powers, and left-to-right evaluation.
• It connects directly to fractions, surds, and indices, since fraction bars and root symbols are grouping devices rather than decoration. A strong understanding here makes later work with rational expressions and scientific notation more reliable.
• It supports equation solving and function evaluation because both depend on interpreting expressions consistently. When substituting into formulas such as $A = \pi r^2$ or evaluating $f(x) = 2x^2 - 3x + 1$, BIDMAS/BODMAS ensures every term is processed in the intended order.
• It also builds mathematical communication, since properly written notation relies on these conventions being shared. Without order of operations, formulas in science, finance, and statistics would need excessive brackets everywhere just to remain unambiguous.