Why We Needed Rules in the First Place
Before standardized notation, even professional mathematicians occasionally disagreed on how to read shared formulas. Once algebra started appearing in printed textbooks, that kind of ambiguity had to go. The rules we call PEMDAS (used in the US) and BODMAS (used in the UK, India, Australia, and elsewhere) are the same underlying priority system - just two different memory aids for it.
| Acronym | Stands For |
|---|---|
| PEMDAS | Parentheses, Exponents, Multiplication, Division, Addition, Subtraction |
| BODMAS | Brackets, Orders, Division, Multiplication, Addition, Subtraction |
The apparent difference - M before D in PEMDAS versus D before M in BODMAS - is misleading. Both systems treat multiplication and division as equal-priority partners. Neither one beats the other. The acronyms just list them in different orders, which causes no end of confusion for students who take the letter sequence literally.
The Four-Tier Priority System
Think of it as four tiers. Operations in a higher tier get handled first, no matter where they appear in the expression.
| Tier | Operation | Notes |
|---|---|---|
| 1 (highest) | Parentheses / Brackets | Any grouping symbols - round ( ), square [ ], curly { } |
| 2 | Exponents / Orders / Roots | Powers, square roots, cube roots |
| 3 | Multiplication and Division | Equal priority - resolved left to right |
| 4 (lowest) | Addition and Subtraction | Equal priority - resolved left to right |
Back to the opening example: 6 + 4 × 2. Multiplication is tier 3, addition is tier 4. Tier 3 goes first: 4 × 2 = 8, then 6 + 8 = 14. The answer 20 is wrong because it processes addition before multiplication.
Step 1 - Brackets: (4 × 2) = 8 → 3 + 5² − 8
Step 2 - Exponent: 5² = 25 → 3 + 25 − 8
Step 3 - No multiplication or division
Step 4 - Addition/Subtraction left to right: 3 + 25 = 28, then 28 − 8 = 20
Same Priority? Go Left to Right.
Tier 3 and tier 4 each have two operations at identical priority. When they show up together in the same expression, you just work left to right - whichever operation comes first, you do first. It's not a tiebreaker rule, it's the actual rule.
Left-to-right: 24 ÷ 6 = 4, then 4 × 2 = 8
Wrong approach - multiplication "first": 6 × 2 = 12, then 24 ÷ 12 = 2 ✗
Left-to-right: 15 − 4 = 11, then 11 + 7 = 18
Wrong approach - addition "first": 4 + 7 = 11, then 15 − 11 = 4 ✗
The one that trips everyone up is division and multiplication. People think multiplication always goes first. It doesn't. The letter order in PEMDAS is just a memory trick, not a ranking. M and D share a tier, and left to right is how you break the tie.
Nested Brackets: Always Start from the Deepest Level
Brackets can be placed inside other brackets. The rule is simple: always resolve the innermost set first, then work your way outward layer by layer. Mathematicians sometimes switch between ( ), [ ], and { } just to make nesting visually clearer, not because the different styles have different priority.
Innermost: (4 − 1) = 3 → 2 × {5 + [3 × 3]}
Next layer: [3 × 3] = 9 → 2 × {5 + 9}
Outer brackets: {5 + 9} = 14 → 2 × 14
Multiply: 2 × 14 = 28
The classic mistake is working from the outside in. Don't. Start at the deepest level and peel outward. Writing each step on its own line, as shown above, makes this a lot easier to track and prevents you from losing your place.
The Mistakes People Actually Make
The following examples are built specifically around the most common misunderstandings. Honestly, if you work through all four of these and understand why the wrong answers are wrong, you're in good shape.
Example A - Negative base vs. negative exponent:
−3² means −(3²) = −9 ← the exponent applies only to 3
(−3)² means (−3) × (−3) = +9 ← the brackets force negation before squaring
Example B - The fraction bar as an implicit bracket:
= 12 ÷ 3 = 4
If you drop the brackets and write 8 + 4 ÷ 2 + 1:
Division first: 4 ÷ 2 = 2 → 8 + 2 + 1 = 11 ✗
Fraction bars in handwritten math act as built-in brackets for numerator and denominator.
Example C - Chained mixed operations:
Step 1 - Exponent: 3² = 9 → 18 ÷ 9 + 6 × 2 − 5
Step 2 - Mult/Div left to right: 18 ÷ 9 = 2, then 6 × 2 = 12 → 2 + 12 − 5
Step 3 - Add/Sub left to right: 2 + 12 = 14, then 14 − 5 = 9
Example D - Consecutive division:
Left-to-right: 100 ÷ 10 = 10, then 10 ÷ 5 = 2
Wrong - dividing by the product: 100 ÷ (10 × 5) = 100 ÷ 50 = 2 coincidentally works here,
but 100 ÷ 10 ÷ 2 gives 5 left-to-right vs. 100 ÷ 20 = 5 - same result only by luck.
Stick to the left-to-right rule; don't collapse divisors unless you add brackets intentionally.
For checking complex expressions, the CalcSolver Pro calculator evaluates the full expression using the correct order of operations before displaying a result - handy for catching arithmetic slips in multi-step problems.