Order of Operations (PEMDAS / BODMAS)
Without a standard order of operations, the expression 2 + 3 × 4 could equal 20 (if you add first) or 14 (if you multiply first). Mathematicians solved this ambiguity centuries ago with a universal agreement — now called PEMDAS in the United States and BODMAS or BEDMAS in many other countries.
This guide walks through every rule, common traps, nested parentheses, and gives you 10 practice problems with full worked solutions. Whether you're a student preparing for a test or refreshing your arithmetic, the order of operations is the single most important rule to get right before tackling algebra.
What Is PEMDAS?
PEMDAS is a mnemonic for the six levels of mathematical priority — the sequence you must follow when evaluating any arithmetic or algebraic expression: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Think of it as a priority queue: operations at the top of the list must be completed before those lower down. When two operations share the same priority level (multiplication and division share level 3; addition and subtraction share level 4), you work left to right — just as you read English text.
The phrase most students use to remember the order is: "Please Excuse My Dear Aunt Sally" — the first letter of each word matches P-E-M-D-A-S.
| Priority | Operation | PEMDAS Letter | Direction |
|---|---|---|---|
| 1 (highest) | Parentheses (and brackets, braces) | P | Inside → out |
| 2 | Exponents (and roots, which are fractional exponents) | E | Right to left* |
| 3 | Multiplication | M | Left to right |
| 3 | Division | D | Left to right (same level as ×) |
| 4 | Addition | A | Left to right |
| 4 | Subtraction | S | Left to right (same level as +) |
PEMDAS vs BODMAS vs BEDMAS — International Differences
The same rule has different names around the world, but the underlying mathematics is identical. The differences are purely mnemonic:
PEMDAS (United States): Parentheses → Exponents → Multiplication → Division → Addition → Subtraction.
BODMAS (UK, India, Australia): Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction. Note: Division appears before Multiplication here, but they remain the same priority level — both evaluated left to right.
BEDMAS (Canada): Brackets → Exponents → Division → Multiplication → Addition → Subtraction. Same rule as BODMAS, different name.
BIDMAS (UK alternative): Brackets → Indices → Division → Multiplication → Addition → Subtraction. "Indices" means exponents/powers.
The most critical point: no matter which acronym you use, the result of any correctly worked expression is the same. The apparent difference between "Multiplication before Division" (PEMDAS) and "Division before Multiplication" (BODMAS) is irrelevant — they're on the same priority level and resolved left to right.
| Acronym | Country | P/B | E/O/I | M&D | A&S |
|---|---|---|---|---|---|
| PEMDAS | USA | Parentheses | Exponents | M then D (left→right) | A then S (left→right) |
| BODMAS | UK/India/AU | Brackets | Orders | D then M (left→right) | A then S (left→right) |
| BEDMAS | Canada | Brackets | Exponents | D then M (left→right) | A then S (left→right) |
| BIDMAS | UK alt. | Brackets | Indices | D then M (left→right) | A then S (left→right) |
Step-by-Step: Applying PEMDAS with Worked Examples
Example 1 — Basic mixed operations: Evaluate 2 + 3 × 4.
Step 1 (Parentheses): None. Step 2 (Exponents): None. Step 3 (Multiplication): 3 × 4 = 12. Rewrite: 2 + 12. Step 4 (Addition): 2 + 12 = 14.
Example 2 — With exponent: Evaluate 3 + 2² × 5 − 1.
Step 1 (P): None. Step 2 (E): 2² = 4. Rewrite: 3 + 4 × 5 − 1. Step 3 (M/D): 4 × 5 = 20. Rewrite: 3 + 20 − 1. Step 4 (A/S): Left to right → 3 + 20 = 23 → 23 − 1 = 22.
Example 3 — Parentheses change everything: Compare 8 ÷ 2(2 + 2) vs (8 ÷ 2)(2 + 2).
For 8 ÷ 2(2 + 2): P first → 2 + 2 = 4. Left to right: 8 ÷ 2 = 4, then 4 × 4 = 16. (This famous internet debate resolves to 16 with standard PEMDAS.) For (8 ÷ 2)(2 + 2): Both parentheses first: 4 × 4 = 16. Same result in this case, but parentheses notation makes the intent unambiguous.
Example 4 — Three-level expression: Evaluate (3 + 5)² ÷ 4 × 2 − 6.
Step 1 (P): 3 + 5 = 8. Rewrite: 8² ÷ 4 × 2 − 6. Step 2 (E): 8² = 64. Rewrite: 64 ÷ 4 × 2 − 6. Step 3 (M/D left to right): 64 ÷ 4 = 16 → 16 × 2 = 32. Rewrite: 32 − 6. Step 4 (S): 32 − 6 = 26.
Nested Parentheses — Working from the Inside Out
When parentheses appear inside other parentheses, always resolve the innermost group first, then work outward. Different bracket types — ( ), [ ], { } — are often used for visual clarity but carry identical mathematical meaning.
Example: Evaluate 3 × {2 + [4 × (1 + 3)]}.
Step 1 — Innermost parentheses: (1 + 3) = 4. Rewrite: 3 × {2 + [4 × 4]}.
Step 2 — Square brackets: [4 × 4] = 16. Rewrite: 3 × {2 + 16}.
Step 3 — Curly braces: {2 + 16} = 18. Rewrite: 3 × 18.
Step 4 — Multiplication: 3 × 18 = 54.
A useful habit: underline the innermost parentheses, compute, rewrite, repeat. Never try to resolve multiple levels in one step — this is where most errors occur.
Exponents and Roots — Common Traps
Trap 1 — Negative base vs negative exponent: −3² ≠ (−3)². Without parentheses, the exponent applies only to 3: −3² = −(3²) = −9. With parentheses: (−3)² = (−3)(−3) = +9. This is one of the most common sign errors in algebra.
Trap 2 — Roots are exponents: √9 is the same as 9^(1/2). Square roots obey the same priority level as exponents — they come before multiplication and division. So 2 × √16 + 1: first √16 = 4, then 2 × 4 = 8, then 8 + 1 = 9.
Trap 3 — Stacked exponents (right-to-left): 2^3^2 means 2^(3²) = 2^9 = 512, not (2³)² = 64. Stacked exponents are evaluated right to left. This is rarely tested at secondary level but matters in computer science.
Trap 4 — Exponent over a fraction: The fraction bar acts as a grouping symbol (like parentheses). In (3 + 5)² / (2 + 2), compute numerator and denominator separately before dividing: 64 / 4 = 16.
Multiplication & Division — Why Left to Right Matters
Many students believe "Multiplication always comes before Division" because M appears before D in PEMDAS. This is a dangerous misreading. Multiplication and division are equal-priority operations — you evaluate them left to right as they appear.
Example showing the difference: 12 ÷ 4 × 3.
Correct (left to right): 12 ÷ 4 = 3, then 3 × 3 = 9.
Incorrect (multiplication first): 4 × 3 = 12, then 12 ÷ 12 = 1. Wrong!
Another example: 100 ÷ 5 ÷ 2. Correct: 100 ÷ 5 = 20, then 20 ÷ 2 = 10. (Not 100 ÷ 10 = 10 — same answer here, but not always.)
The safest habit: whenever you see a chain of × and ÷, scan left to right and process each operation in the order it appears.
Addition & Subtraction — Left to Right (Same Priority)
Just like multiplication and division, addition and subtraction share the same priority level. Work left to right.
Example: 10 − 3 + 2. Correct: 10 − 3 = 7, then 7 + 2 = 9. Incorrect (addition first): 3 + 2 = 5, then 10 − 5 = 5. Wrong!
The reason this matters: subtraction is not commutative or associative. a − b + c ≠ a − (b + c) in general. Always process left to right.
Dealing with negative signs: Think of subtraction as "adding a negative number." So 8 − 5 + 3 can be rewritten as 8 + (−5) + 3. Then order doesn't matter — but this reformulation only works if you're careful to carry the negative sign with its number.
8 Common PEMDAS Mistakes (and How to Avoid Them)
Understanding the rules is one thing; applying them under pressure is another. These are the most frequent errors at every level.
| Mistake | Wrong | Correct | Why |
|---|---|---|---|
| Multiplication before division (left to right) | 12 ÷ 4 × 3 = 1 | 12 ÷ 4 × 3 = 9 | M and D are equal priority — go left to right |
| Addition before subtraction (left to right) | 10 − 3 + 2 = 5 | 10 − 3 + 2 = 9 | A and S are equal priority — go left to right |
| Forgetting exponent sign | −3² = 9 | −3² = −9 | Exponent binds to 3, not −3 (use parentheses for −3²=9) |
| Ignoring fraction bar as grouping | (3+5)/2+2 = 6 | (3+5)/(2+2) = 2 | Fraction bar groups numerator AND denominator |
| Skipping steps in nested brackets | {2+[3×(1+1)]} | Inner first: (1+1)=2 → [3×2]=6 → {2+6}=8 | Always innermost first |
| Implicit multiplication after parentheses | 8÷2(4) = 1 | 8÷2×4 = 16 | Implicit × has same priority as ÷ — go left to right |
| Root not treated as exponent | √4+5 = 3 | √4+5 = 7 | √4=2 first (exponent level), then 2+5=7 |
| Applying PEMDAS to equations (both sides) | Wrong variable isolation | Use inverse operations side by side | PEMDAS is for evaluating expressions, not solving equations |
PEMDAS in Algebra — Variables and Expressions
PEMDAS applies to algebraic expressions exactly as it does to numeric ones. The only difference is that you can't always complete the arithmetic when variables are present — but you follow the same evaluation order.
Example: Simplify 3x² + 2(x + 4) − 5x when x = 2.
Step 1: Substitute x = 2 → 3(2²) + 2(2 + 4) − 5(2).
Step 2 (Parentheses): (2 + 4) = 6 → 3(4) + 2(6) − 5(2).
Step 3 (Exponents): 2² = 4 (already done above).
Step 4 (Multiplication): 3 × 4 = 12, 2 × 6 = 12, 5 × 2 = 10 → 12 + 12 − 10.
Step 5 (A/S left to right): 12 + 12 = 24 → 24 − 10 = 14.
When simplifying without substitution, combine like terms after expanding: distribute first (respecting multiplication), then collect like-degree terms. The factoring calculator and polynomial calculator on CalcMulti handle these expansions automatically.
10 Practice Problems with Full Solutions
Problem 1: 5 + 3 × 2 − 1 = ? Solution: 3×2=6 → 5+6−1=10.
Problem 2: (5 + 3) × 2 − 1 = ? Solution: (5+3)=8 → 8×2=16 → 16−1=15.
Problem 3: 4² − 2 × (3 + 1) = ? Solution: (3+1)=4 → 4²=16 → 2×4=8 → 16−8=8.
Problem 4: 3 × 4 ÷ 2 + 1 = ? Solution: left to right → 3×4=12 → 12÷2=6 → 6+1=7.
Problem 5: 2 + {3 × [4 − (1 + 1)]} = ? Solution: (1+1)=2 → [4−2]=2 → {3×2}=6 → 2+6=8.
Problem 6: (−2)³ + 4 × 2 = ? Solution: (−2)³=−8 → 4×2=8 → −8+8=0.
Problem 7: 18 ÷ 3 ÷ 2 × 4 = ? Solution: left to right → 18÷3=6 → 6÷2=3 → 3×4=12.
Problem 8: √(16 + 9) × 2 = ? Solution: (16+9)=25 → √25=5 → 5×2=10.
Problem 9: 100 − 4 × (2 + 3)² ÷ 5 = ? Solution: (2+3)=5 → 5²=25 → 4×25=100 → 100÷5=20 → 100−20=80.
Problem 10: 2³ × 3 − (4 + 1)² ÷ 5 + 7 = ? Solution: 2³=8, (4+1)=5, 5²=25 → 8×3=24 → 25÷5=5 → 24−5+7=26.
How Calculators Handle Order of Operations
Not all calculators follow PEMDAS automatically. Basic four-function calculators (the cheap ones) evaluate left to right — they have no memory for priority. 2 + 3 × 4 gives 20, not 14, on a basic calculator.
Scientific calculators (Casio, Texas Instruments) and all smartphone calculators (iOS, Android) implement the full order of operations. They are safe to use for multi-step expressions.
Spreadsheets (Excel, Google Sheets) also follow standard order of operations. However, the caret ^ for exponents and the placement of negative signs can cause surprises — always use parentheses to be explicit: =(3+5)^2/(2+2).
Programming languages: Python, JavaScript, and most languages follow standard mathematical precedence. However, integer division behaves differently (e.g., 7 // 2 = 3 in Python). When in doubt, use parentheses — they cost nothing and prevent bugs.
The safest rule when using any tool: write your own parentheses explicitly for each intended grouping, rather than relying on operator precedence rules you're uncertain about.