Order of Operations (PEMDAS / BODMAS)
| Tier | Operation & Symbols | Key Rule | Quick Example |
|---|---|---|---|
| 1 | Parentheses / Grouping ( ) [ ] | Solve innermost brackets first; top/bottom of fraction bars | (4 + 3) ร 2 = 7 ร 2 = 14 |
| 2 | Exponents & Radicals (xยฒ, โx) | Evaluate powers, squares, and square roots | 2 ร 3ยฒ = 2 ร 9 = 18 |
| 3 | Multiply & Divide (ร, รท) | EQUAL PRIORITY โ Solve strictly LEFT TO RIGHT | 12 รท 3 ร 2 = 4 ร 2 = 8 |
| 4 | Add & Subtract (+, -) | EQUAL PRIORITY โ Solve strictly LEFT TO RIGHT | 20 - 4 + 6 = 16 + 6 = 22 |
20 - 4 + 6 ร 2โข Mult first: 6 ร 2 = 12 โ
20 - 4 + 12โข Left-to-Right: 20 - 4 = 16 โ 16 + 12 = 28
Problem 2:
3 ร (4 + 2)ยฒ รท 4โข Parentheses: 4 + 2 = 6 โ
3 ร 6ยฒ รท 4โข Exponent: 6ยฒ = 36 โ
3 ร 36 รท 4โข Left-to-Right: 3 ร 36 = 108 โ 108 รท 4 = 27
Problem 3:
45 - [3 ร (2ยณ + 1) - 7]โข Inside ( ): 2ยณ = 8 โ 8 + 1 = 9 โ
45 - [3 ร 9 - 7]โข Inside [ ]: 3 ร 9 = 27 โ 27 - 7 = 20 โ
45 - 20 = 2518 รท 3 ร 2, do NOT multiply 3 ร 2 first! Division is on the left, so 18 รท 3 = 6, then 6 ร 2 = 12.โข Trap 2 (Add vs Sub): In
10 - 3 + 2, do NOT add 3 + 2 first! Subtraction is on the left, so 10 - 3 = 7, then 7 + 2 = 9.โข Trap 3 (Fraction Bar): The horizontal bar acts as grouping!
(8 + 4) / (2 ร 3) = 12 / 6 = 2.โข Trap 4 (Negative Base):
-3ยฒ = -(3 ร 3) = -9, whereas (-3)ยฒ = +9.Order of Operations: PEMDAS, BODMAS & The Left-to-Right Rule
Why We Can't Read Math Like a Book (The Airport VIP Rule)
When you read a sentence in English, you read strictly from left to right. But mathematics is like an airport boarding gate: some passengers have higher priority passes and board the plane first!
If everyone rushed in at once, a simple expression like 4 + 3 ร 5 would cause chaos. One person would say 35 (adding first), and another would say 19 (multiplying first).
PEMDAS vs BODMAS vs GEMDAS: What's the Difference?
Depending on where you go to school in the world, your teacher might use a different mnemonic memory phrase. They all mean the exact same mathematical rules!
โข E: Exponents
โข M & D: Multiply & Divide (Left to Right)
โข A & S: Add & Subtract (Left to Right)
"Please Excuse My Dear Aunt Sally"
โข O: Orders (Powers / Roots)
โข D & M: Divide & Multiply (Left to Right)
โข A & S: Add & Subtract (Left to Right)
Standard British Commonwealth curriculum
โข E: Exponents & Radicals
โข MD: Multiplication / Division (Equal Rank)
โข AS: Addition / Subtraction (Equal Rank)
Preferred by modern mathematicians!
The 4 Traps Where 80% of Students Lose Marks
In PEMDAS, M is written before D, but they have 100% equal priority. When both appear, you MUST solve from Left to Right:
โ Common Mistake: 12 รท (3 ร 2) = 12 รท 6 = 2 (WRONG!)
Just like multiplication and division, Addition and Subtraction are equal partners. Solve from Left to Right:
โ Common Mistake: 20 - (4 + 6) = 20 - 10 = 10 (WRONG!)
A horizontal fraction bar acts like giant invisible parentheses around the top and bottom:
-3ยฒ means -(3 ร 3) = -9 (Exponent applies ONLY to 3, then negative is applied).โข
(-3)ยฒ means (-3) ร (-3) = +9 (Parentheses protect the negative sign).
PDF + Answer KeysOrder of Operations & Integers 50-Worksheet Master Workbook
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3 Step-by-Step Worked Examples
Master how to systematically solve every tier of expressions:
6 ร 2 = 12 โ 20 - 4 + 1220 - 4 = 16 โ 16 + 1216 + 12 = 28.4 + 2 = 6 โ 3 ร 6ยฒ รท 46ยฒ = 36 โ 3 ร 36 รท 43 ร 36 = 108 โ 108 รท 4 = 27.2ยณ = 8 โ (8 + 1) = 9 โ 45 - [3 ร 9 - 7]3 ร 9 = 27 โ 27 - 7 = 20 โ 45 - [20]45 - 20 = 25.๐ฏ Active Recall Quiz (Test Your PEMDAS Mastery)
Test your understanding of tiebreakers and parentheses:
Frequently Asked Questions
The order of operations is the universal rulebook for solving math expressions: 1) Parentheses / Grouping Symbols, 2) Exponents / Powers, 3) Multiplication and Division (from left to right), and 4) Addition and Subtraction (from left to right).
No! This is the #1 most common student mistake. Multiplication and Division have EQUAL priority. When both appear in an expression, you must solve them in order from LEFT to RIGHT.
PEMDAS and BODMAS represent the exact same mathematical rules. PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is used primarily in the US and Canada. BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is used in the UK, India, Australia, and New Zealand.
Without a standardized order of operations, a simple expression like 4 + 3 ร 5 could produce two different answers (35 if adding first, or 19 if multiplying first). The order of operations ensures every mathematician, student, and computer calculates the exact same correct answer (19).
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