The expression 3 + 4 × 2 has exactly one correct value, 11, even though reading it left to right gives 14. That gap between reading order and computing order is the whole subject of this guide. Mathematics needed a shared convention so that an expression written in one classroom means the same thing in every other, and the convention it chose is the order of operations: brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. This guide teaches that order with worked evaluations, shows how it connects to solving equations — every substitution check is an expression evaluation — and names the mistakes that most often break it. One honesty note first: the Math Solver on this site takes coefficients, not free-form expressions, so there is no box where you type 3 + 4 × 2. You evaluate expressions by hand, using this order, and you use the solver for equations.

The order, stated once and used everywhere

The order of operations is a reading convention, not a law of nature. An expression such as 3 + 4 × 2 is ambiguous in plain English — it could mean “add, then multiply” or “multiply, then add” — so mathematics fixed one reading and taught it everywhere. The reading is: brackets first, then powers (exponents), then multiplication and division, then addition and subtraction. Two common memory aids encode it: PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) and BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). They describe the same order, and the apparent disagreement between them — PEMDAS lists multiplication before division, BODMAS lists division before multiplication — is not a disagreement at all, as the section on equal pairs explains. What matters is not the mnemonic but the habit: before computing anything, scan the whole expression and mark the order in which its operations will happen.

Why this particular order? Because it preserves the way expressions are built. Multiplication is repeated addition, so 4 × 2 is a single quantity, a block, in a way that 3 + 4 is not: 3 + 4 × 2 means 3 plus the block (4 × 2), which is 3 + 8 = 11. Powers are repeated multiplication, so 2 × 3² means 2 times the block (3²), which is 2 × 9 = 18, not 36. Brackets are the escape hatch: anything inside brackets becomes a block first, whatever it contains, so (3 + 4) × 2 means the block (3 + 4) times 2, which is 14. Once you see operations as blocks with a priority ranking, the order stops feeling arbitrary and starts feeling like grammar: brackets are the innermost clause, powers bind tighter than products, products bind tighter than sums.

Brackets: the override that changes answers

Brackets outrank everything. When an expression contains brackets, evaluate the innermost brackets first, replace them with their value, and then continue with the rest of the expression as if the brackets had never been there. For 5 + 2 × (8 − 3)², the bracket (8 − 3) gives 5, the power gives 25, the multiplication gives 50, and the addition gives 55. Skip the bracket step and the answer collapses: 5 + 2 × 8 − 3² evaluated in the standard order is 5 + 16 − 9 = 12, a different expression with a different answer, and no partial credit for “almost the same”. The lesson is mechanical: brackets are not decoration. They are instructions, and they are obeyed before anything else.

Nested brackets follow the same rule recursively: work from the inside out. For 2 × ((3 + 1) × (5 − 2)), the innermost brackets give 4 and 3, so the expression becomes 2 × (4 × 3) = 2 × 12 = 24. A useful physical habit is to rewrite the expression after each step, crossing out the part you have evaluated and writing its value in its place. Students who try to hold intermediate values in their heads are the ones who drop a sign or lose a bracket; students who rewrite, step by step, have a written trail they can check. The solver’s own step listings work the same way — each step shows the equation in its new form, so any error can be located to a single line — and your handwritten working should do the same.

Brackets also appear invisibly, and recognising them is a skill in its own right. A fraction bar groups its numerator and denominator: (10 − 6) ÷ 2 is 2, while 10 − 6 ÷ 2 is 7, because the bar in (10 − 6)/2 acts as brackets around the subtraction. A square root sign groups what sits beneath it. And in the quadratic formula, the numerator (−b ± √D) and the denominator (2a) are each bracketed groups: for x² − 5x + 6 = 0, the roots are (5 ± 1) ÷ 2, not 5 ± 1 ÷ 2, which would be 5 ± 0.5. Whenever an operation is written as a fraction, a root, or a formula, draw the invisible brackets in before you compute.

Powers before multiplication

A power binds tighter than multiplication, which means 2 × 3² is 2 × 9 = 18, not 6² = 36. The power belongs to the number immediately beneath it — the base — and nothing else. That single rule explains the most common power error: squaring a product instead of a factor. (2 × 3)² is 36, because the brackets make the product the base, but 2 × 3² is 18, because only the 3 is squared. Similarly, −3² is −9, because the power binds to 3 and the minus sign sits outside it, while (−3)² is 9, because the brackets make −3 the base. Examiners test this distinction constantly, and it is worth practising until it is automatic: ask, before squaring anything, “what exactly is the base?”

Powers also interact with the other operations in ways that reward a slow first pass. For 4 + 3² × 2, the order is: power (9), multiplication (18), addition (22). For 3² + 4², both powers come before the addition: 9 + 16 = 25. For (3 + 4)², the bracket comes first: 7² = 49, which is not 3² + 4² = 25 — squaring a sum is not the sum of the squares, a mistake so common it deserves its own warning. And a power of a power, such as (2³)², is handled inside out: 2³ = 8, then 8² = 64, which agrees with the exponent rule 2⁶ = 64. When in doubt, expand: 3² means 3 × 3, and writing the expansion out removes all doubt about what is being multiplied.

The two equal pairs, and left-to-right

Multiplication and division have equal priority, and addition and subtraction have equal priority. Within each pair, work from left to right. This is where both mnemonics mislead beginners: PEMDAS seems to say multiplication always precedes division, and BODMAS seems to say the reverse, but neither is true. For 12 ÷ 3 × 2, left to right gives (12 ÷ 3) × 2 = 8. Doing the multiplication first would give 12 ÷ 6 = 2, which is wrong. The pair rule also explains subtraction chains: 10 − 6 + 2, left to right, is 4 + 2 = 6, while adding first would give 10 − 8 = 2, which is wrong. Division and subtraction are not “done last”; they are done in turn, in position.

A cleaner way to think about the second pair is that subtraction is addition of a negative: 10 − 6 + 2 is 10 + (−6) + 2, and addition in any order gives 6. Similarly, division is multiplication by a reciprocal: 12 ÷ 3 × 2 is 12 × (1/3) × 2 = 8 regardless of order. These reframings are not just tricks; they are the reason the left-to-right rule exists. Multiplication and division are the same operation in disguise, as are addition and subtraction, so within each pair there is no priority to apply — only position. When an expression mixes the pairs, the higher pair still wins: in 10 − 6 ÷ 2, the division (priority 3) happens before the subtraction (priority 4), giving 10 − 3 = 7, and no left-to-right reading can override that.

Negatives, fractions, and decimals

Negative numbers obey the same order, but they punish carelessness more visibly. For −2 × 3 + 5, the multiplication comes first: −6 + 5 = −1. For 5 − (−3), subtracting a negative is adding a positive: 5 + 3 = 8. For −3², as noted above, the answer is −9, while (−3)² is 9. The safest habit with negatives is to bracket them the moment they appear: rewrite −3² as −(3²) and (−3)² as it stands, and the difference becomes visible rather than a matter of memory. The same habit helps with the solver’s coefficient boxes, where a coefficient of −5 must be entered with its sign attached to the number, not left floating as a separate subtraction.

Fractions and decimals follow the same order, with one extra caution: a fraction bar is a bracket. For 1/2 × 8 + 3, the multiplication comes first: 4 + 3 = 7. For (1 + 2) / (3 − 1), the numerator and denominator are each evaluated first: 3 / 2 = 1.5. Decimals behave identically: 0.5 × 8 + 3 = 7, and 2.5² = 6.25, not 5. The connection to the solver matters here: its coefficient boxes accept decimals such as 0.5 but not a slash, so a coefficient of 1/2 must be entered as 0.5 — and the solver’s exact-fraction reporting then gives you back the fraction form of the answer, such as x = 1/2, where the arithmetic allows it. Entering the decimal is a change of notation, not a change of value, provided you convert correctly: 1/2 = 0.5, 1/4 = 0.25, 1/3 = 0.333… with the last one necessarily approximate, which is exactly why exact fractions are kept until the final line.

Where order of operations meets equation solving

Every substitution check is an expression evaluation, which makes order of operations the skill underneath equation solving. Checking x = 3 in x² − 5x + 6 = 0 means evaluating 3² − 5 × 3 + 6: power first (9), multiplication (15), then left-to-right addition and subtraction (9 − 15 + 6 = 0). Evaluate it left to right instead — 3² is still 9, but a careless reader might compute 9 − 5 = 4, then 4 × 3 = 12, then 12 + 6 = 18 — and a correct root appears to fail its check, which is worse than no check at all, because it teaches distrust of a right answer. The solver’s check lines show the evaluation in full, with each term separated, precisely so you can follow the order: 9 − 15 + 6 = 0 is three terms combined, not a chain of operations.

The same skill appears inside the quadratic formula itself. Computing D = b² − 4ac for a = 1, b = −5, c = 6 means: square b (25), multiply 4 × a × c (24), subtract (1). The multiplication 4ac is a single block, evaluated before the subtraction, even though it is written after the b² term. And the roots (5 ± 1) ÷ 2 require the bracket first: the numerator is a complete block before the division happens. Students who can evaluate expressions reliably rarely struggle with the formula’s arithmetic; students who cannot, struggle with the formula even when they understand it perfectly. If your quadratic answers keep failing their checks, practise expression evaluation before practising quadratics — the fault is usually upstream.

The mistakes that break evaluations

The mistakes fall into a short, recognisable list. First, reading left to right: 3 + 4 × 2 computed as 14. Second, the false mnemonic: doing all multiplications before all divisions, or all additions before all subtractions, instead of left to right within each equal pair — 12 ÷ 3 × 2 computed as 2. Third, squaring the wrong base: 2 × 3² computed as 36, or −3² computed as 9. Fourth, distributing a power over a sum: (3 + 4)² computed as 25. Fifth, ignoring invisible brackets: a fraction bar or a formula’s numerator treated as separate operations, so (10 − 6) ÷ 2 becomes 7. Sixth, dropping a bracket mid-calculation: rewriting (8 − 3)² as 8 − 3², or losing a minus sign when a bracket is removed. Every one of these is a reading error, not a computing error, and every one is caught by the same defence: rewrite the expression after each step, with the evaluated part replaced by its value, and check the final value by estimating — 3 + 4 × 2 must be near 11, not near 14, because 4 × 2 is only 8.

The defence that matters most is also the simplest: slow down on the first pass. Before computing anything, scan the expression and number its operations in order — brackets 1, powers 2, products and quotients 3, sums and differences 4 — and then execute in that order, rewriting as you go. It feels slow at first, and then it becomes fast, because the errors it prevents are the ones that cost the most time to find later. An expression evaluated correctly the first time needs no rework; an expression evaluated quickly and wrongly needs to be solved twice, once wrongly and once rightly, with the added risk that the wrong value is carried into the next line. Order of operations is not a topic to pass and forget. It is the grammar of every calculation that follows, and fluency in it is what makes harder mathematics feel easier.

Frequently asked questions

What is the order of operations?

Brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Memory aids are PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) and BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction); both describe the same order, because multiplication and division are an equal pair, as are addition and subtraction.

Why is 3 + 4 × 2 equal to 11 and not 14?

Because multiplication outranks addition: 4 × 2 is a single block equal to 8, and 3 + 8 = 11. Reading left to right gives (3 + 4) × 2 = 14, which is a different expression — the brackets in (3 + 4) × 2 are what force the addition first. Brackets are the only way to override the standard order.

Does multiplication always come before division?

No. Multiplication and division have equal priority, so within a chain of multiplications and divisions you work left to right: 12 ÷ 3 × 2 = 8, not 2. The same applies to addition and subtraction: 10 − 6 + 2 = 6, not 2. The mnemonics list one operation first only as a naming convenience, not as a priority rule.

What is the difference between −3² and (−3)²?

−3² is −9, because the power binds only to 3 and the minus sign sits outside it: −(3²) = −9. (−3)² is 9, because the brackets make −3 itself the base: (−3) × (−3) = 9. Always ask what exactly the base is before squaring anything.

How does order of operations relate to solving equations?

Every substitution check is an expression evaluation. Checking x = 3 in x² − 5x + 6 = 0 means evaluating 9 − 15 + 6 in the correct order, giving 0. A check evaluated in the wrong order can make a correct root look wrong, so expression fluency is a prerequisite for reliable equation solving.

Can the Math Solver evaluate expressions like 3 + 4 × 2?

No. The solver on this site takes coefficients, not free-form expressions: its boxes hold the numbers in ax + b = c or ax² + bx + c = 0. Evaluate expressions by hand using the order in this guide, and use the solver for equations — its substitution checks then show expression evaluation applied to your roots.