What Is PEMDAS? Rules, Examples, and Order of Operations Guide
Have you ever gotten a different answer than your classmate when solving the same math problem? Or wondered why your calculator gives one result while doing it "left to right" gives another? The confusion usually comes down to one thing: not following the correct order of operations. That's exactly where PEMDAS comes in.
In this comprehensive guide, you'll discover exactly what PEMDAS stands for, when and how to apply it correctly, common mistakes that trip students up, and see multiple worked examples from simple to complex. We'll cover the detailed rules, explore variations like BODMAS and BEDMAS, tackle frequently asked questions, and provide you with a handy cheat sheet you can reference anytime. Whether you're a student struggling with homework, a parent helping with assignments, or just refreshing your math skills, this guide will help you master the order of operations once and for all.
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What Is PEMDAS? — Definition & Meaning
PEMDAS is an acronym that tells you the correct order to solve arithmetic expressions involving multiple operations: P for Parentheses first, then E for Exponents (powers and roots), then M and D for Multiplication and Division (left to right), and finally A and S for Addition and Subtraction (left to right). Following PEMDAS ensures you get the same correct result no matter who solves the problem.
PEMDAS Full Form
PEMDAS stands for:
P – Parentheses (or any grouping symbols like brackets [ ] or braces { })
E – Exponents (including powers, roots, and indices)
M – Multiplication
D – Division
A – Addition
S – Subtraction
The most popular mnemonic to remember this sequence is "Please Excuse My Dear Aunt Sally," where each word's first letter corresponds to the operation order.
Why PEMDAS Exists — The Need for an Order of Operations
Without a standardized order of operations, mathematical expressions would be ambiguous. Consider the simple expression: 6 + 3 × 2
If you work left to right, you'd get: 6 + 3 = 9, then 9 × 2 = 18
But following PEMDAS (multiplication before addition), you get: 3 × 2 = 6, then 6 + 6 = 12
The correct answer is 12. PEMDAS exists to eliminate this confusion and ensure mathematicians, students, calculators, and computers all interpret expressions the same way. It's a universal convention that makes mathematical communication clear and consistent.
PEMDAS Rule — Step-by-Step Order of Operations
Understanding the hierarchy of operations is essential for solving expressions correctly. Here's the detailed breakdown:
The PEMDAS Hierarchy
1. Parentheses (and all grouping symbols) – Always evaluate expressions inside parentheses, brackets, or braces first. If you have nested grouping symbols (parentheses inside parentheses), work from the innermost outward.
2. Exponents – After handling all grouping symbols, evaluate exponents, powers, and roots next.
3. Multiplication and Division – These operations have equal precedence and should be performed from left to right as they appear in the expression.
4. Addition and Subtraction – These also have equal precedence and should be performed from left to right as they appear.
Important Nuance: Left-to-Right Rule
One of the most critical aspects of PEMDAS that often confuses students is this: Multiplication does NOT always come before Division, and Addition does NOT always come before Subtraction.
When operations have the same level of precedence (M and D are equal, A and S are equal), you work through them from left to right. For example:
In 20 ÷ 4 × 5, you do NOT do multiplication first. You work left to right: 20 ÷ 4 = 5, then 5 × 5 = 25.
In 10 − 3 + 2, you do NOT do addition first. You work left to right: 10 − 3 = 7, then 7 + 2 = 9.
Alternative Mnemonics and Variations
Different countries and educational systems use different acronyms for the same concept:
BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) – common in the UK and Commonwealth countries
BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction) – used in Canada
GEMDAS (Grouping, Exponents, Multiplication, Division, Addition, Subtraction) – another variation
Despite the different names, the underlying mathematical principle is identical. The order of operations is a universal mathematical convention.
How to Use PEMDAS — Worked Examples
Let's walk through several examples, progressing from simple to complex, to see PEMDAS in action.
Example 1: Simple Expression
Problem: 5 + 3 × 2
Step-by-step solution:
Check for parentheses – none
Check for exponents – none
Perform multiplication: 3 × 2 = 6
Perform addition: 5 + 6 = 11
Answer: 11
(If you worked left to right incorrectly, you'd get 5 + 3 = 8, then 8 × 2 = 16, which is wrong.)
Example 2: With Parentheses
Problem: (8 + 2) × 5 − 3
Step-by-step solution:
Parentheses first: 8 + 2 = 10
Now we have: 10 × 5 − 3
Multiplication: 10 × 5 = 50
Subtraction: 50 − 3 = 47
Answer: 47
Example 3: Including Exponents
Problem: 4 + 2³ × 3
Step-by-step solution:
Check for parentheses – none
Exponents: 2³ = 8
Now we have: 4 + 8 × 3
Multiplication: 8 × 3 = 24
Addition: 4 + 24 = 28
Answer: 28
Example 4: Division and Multiplication (Left-to-Right)
Problem: 24 ÷ 6 × 2
Step-by-step solution:
No parentheses or exponents
Both division and multiplication present – work left to right
Division first (because it's leftmost): 24 ÷ 6 = 4
Then multiplication: 4 × 2 = 8
Answer: 8
(Common mistake: doing multiplication first would give 6 × 2 = 12, then 24 ÷ 12 = 2, which is incorrect.)
Example 5: Complex with Nested Parentheses
Problem: 3 × [15 − (3 + 2)] + 4²
Step-by-step solution:
Start with innermost parentheses: 3 + 2 = 5
Now we have: 3 × [15 − 5] + 4²
Next grouping symbol (brackets): 15 − 5 = 10
Now we have: 3 × 10 + 4²
Exponents: 4² = 16
Now we have: 3 × 10 + 16
Multiplication: 3 × 10 = 30
Addition: 30 + 16 = 46
Answer: 46
Example 6: All Operations Combined
Problem: 100 − 2³ + (12 ÷ 3) × 5
Step-by-step solution:
Parentheses: 12 ÷ 3 = 4
Now we have: 100 − 2³ + 4 × 5
Exponents: 2³ = 8
Now we have: 100 − 8 + 4 × 5
Multiplication: 4 × 5 = 20
Now we have: 100 − 8 + 20
Work left to right for subtraction and addition: 100 − 8 = 92
Then: 92 + 20 = 112
Answer: 112
Common Mistakes & Misunderstandings with PEMDAS
Even when students know about PEMDAS, certain mistakes keep appearing. Here are the most common pitfalls and how to avoid them:
Mistake 1: Always Doing M Before D or A Before S
Many students memorize PEMDAS and think multiplication ALWAYS comes before division, or addition ALWAYS comes before subtraction. This is incorrect.
The truth: Multiplication and Division have equal priority—work left to right. Addition and Subtraction have equal priority—work left to right.
Example of the error: Expression: 10 − 4 + 2
Wrong approach: "A comes before S, so 4 + 2 = 6, then 10 − 6 = 4" Correct approach: Work left to right → 10 − 4 = 6, then 6 + 2 = 8
Mistake 2: Forgetting About Nested Grouping Symbols
When you have parentheses inside brackets or multiple layers of grouping, you must work from the innermost outward.
Example: Expression: 2 × [8 − (3 + 1)]
You cannot skip the inner parentheses. Solve (3 + 1) = 4 first, then [8 − 4] = 4, then 2 × 4 = 8.
Mistake 3: Ignoring or Forgetting Exponents
Exponents are easy to overlook, especially in longer expressions, but they must be evaluated right after parentheses and before multiplication or division.
Example: Expression: 5 + 3² × 2
Wrong: 5 + 3 = 8, then 8² = 64, then 64 × 2 = 128 Correct: 3² = 9 first, then 9 × 2 = 18, then 5 + 18 = 23
Mistake 4: Confusing Different Acronyms
Students sometimes get confused when they learn BODMAS in one class and PEMDAS in another, thinking they're different rules. They're not—just different names for the same concept. Brackets = Parentheses, Orders = Exponents.
PEMDAS Around the World — Acronyms and Regional Differences
The order of operations is a universal mathematical principle, but different regions teach it using different mnemonics:
PEMDAS (United States) – Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
BODMAS (UK, India, Australia) – Brackets, Orders, Division, Multiplication, Addition, Subtraction
BEDMAS (Canada) – Brackets, Exponents, Division, Multiplication, Addition, Subtraction
BIDMAS (Some UK schools) – Brackets, Indices, Division, Multiplication, Addition, Subtraction
GEMDAS – Grouping, Exponents, Multiplication, Division, Addition, Subtraction
All of these represent the exact same mathematical convention. The key takeaway: regardless of which acronym you learned, the mathematical principle remains identical.
It's worth noting that as you advance in mathematics—particularly in algebra, calculus, and symbolic computation—the strict PEMDAS hierarchy becomes less of a focal point. Mathematical notation itself (like fraction bars, function notation, and matrices) makes the order clear without needing to constantly reference the acronym. However, PEMDAS remains an essential foundation for understanding basic arithmetic and algebra.
When PEMDAS Doesn't Apply / Limitations & What to Know
While PEMDAS is an excellent teaching tool and works perfectly for standard arithmetic and algebraic expressions, it's important to understand its limitations:
Advanced Mathematical Contexts
In higher mathematics, notation often makes grouping and precedence clear without needing PEMDAS. For example:
Fraction bars act as grouping symbols: (a + b)/(c + d) clearly shows the entire numerator and denominator are separate groups
Function notation: f(x + 2) makes it clear that x + 2 is evaluated before applying function f
Matrices and vectors have their own operational rules
Implicit Multiplication Debates
Some mathematicians and educators debate whether implicit multiplication (like 2(3 + 4) without an × sign) should take precedence over explicit division. This has led to viral "calculator arguments" on social media. Most modern calculators and mathematical conventions treat implicit and explicit multiplication the same way, but be aware this ambiguity exists.
Different Operator Types
PEMDAS specifically addresses basic arithmetic operators. When you encounter operations like:
Modulo operations
Matrix multiplication
Set operations
Logical operators
The standard PEMDAS hierarchy may not directly apply, and you'll need to learn the specific precedence rules for those contexts.
PEMDAS as a Learning Tool
Think of PEMDAS as a fundamental principle for arithmetic and basic algebra—not as the "supreme law of all mathematics." It's a convention that allows us to communicate clearly, but mathematical notation, context, and advanced operations can override or extend beyond simple PEMDAS rules.
Quick PEMDAS Cheat Sheet & Summary
Here's a quick reference you can bookmark or print for easy access:
The PEMDAS Order
P – Parentheses (brackets, braces, grouping symbols) – Work from innermost to outermost
E – Exponents (powers, roots, indices)
MD – Multiplication & Division (equal precedence, work LEFT TO RIGHT)
AS – Addition & Subtraction (equal precedence, work LEFT TO RIGHT)
Key Reminders
✓ Multiplication doesn't always come before division ✓ Addition doesn't always come before subtraction ✓ When operations have equal precedence, work left to right ✓ Nested parentheses: work from inside out ✓ Don't skip any steps—work through systematically
Mnemonic Devices
"Please Excuse My Dear Aunt Sally" (PEMDAS)
"Big Old Dogs Make Awful Sounds" (BODMAS)
"Brackets Eliminated, Division, Multiplication, Addition, Subtraction" (BEDMAS)
Quick Self-Test
Can you solve these correctly?
7 + 3 × 4 = ? (Answer: 19)
20 ÷ 5 × 2 = ? (Answer: 8)
(5 + 3)² − 10 = ? (Answer: 54)
FAQs — Frequently Asked Questions About PEMDAS
What does the "E" in PEMDAS stand for?
The "E" stands for Exponents, which includes powers (like 5²), roots (like √9), and any expression with indices. This is the second step in the order of operations, right after handling parentheses.
Is PEMDAS the same as BODMAS or BEDMAS?
Yes, they're the same mathematical concept with different names. PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is used primarily in the United States, while BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is common in the UK and Commonwealth countries. BEDMAS is the Canadian version. All represent the identical order of operations.
What happens if an expression has both multiplication and division? Which comes first?
Neither—they have equal precedence. When you encounter both multiplication and division in an expression, you work through them from left to right as they appear. For example, in 24 ÷ 6 × 2, you divide first (because it's on the left) to get 4, then multiply by 2 to get 8.
Why do we need PEMDAS? Can't we just do operations left to right?
Without PEMDAS, mathematical expressions would be ambiguous and could have multiple interpretations. For instance, 5 + 3 × 2 could equal either 16 (if working strictly left to right) or 11 (following PEMDAS). The order of operations ensures everyone—students, teachers, calculators, and computers—interprets expressions the same way and gets the same correct answer.
Is PEMDAS applicable in algebraic expressions with variables?
Absolutely. PEMDAS applies to all arithmetic and algebraic expressions, whether they contain numbers, variables, or both. For example, in 3x + 2x², you'd evaluate the exponent first (2x²), then handle the addition. The presence of variables doesn't change the order of operations.
What if there are nested parentheses, brackets, or braces?
When you have nested grouping symbols—like parentheses inside brackets inside braces—work from the innermost grouping outward. For example, in {5 × [8 − (3 + 1)]}, you'd first solve (3 + 1) = 4, then [8 − 4] = 4, then {5 × 4} = 20.
Is PEMDAS used in all countries?
The concept of order of operations is universal across all countries and mathematical systems. However, different regions use different acronyms: PEMDAS in the US, BODMAS in the UK and India, BEDMAS in Canada, and so on. The underlying mathematical principle is identical worldwide.
Does PEMDAS apply to calculators and computers?
Yes, scientific calculators and computer programming languages are designed to follow the standard order of operations. However, basic four-function calculators might process operations strictly left to right, which can give incorrect results for complex expressions. This is why it's important to understand PEMDAS even when using technology.
Conclusion
Mastering PEMDAS—the order of operations—is fundamental to solving arithmetic and algebraic expressions correctly. By following the clear hierarchy of Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right), you'll ensure consistent, accurate results every time.
Remember the key nuances: multiplication doesn't automatically trump division, and addition doesn't automatically trump subtraction. When operations share the same precedence level, work through them from left to right. Pay special attention to nested grouping symbols, never skip exponents, and practice with varied examples from simple to complex.
The more you practice applying PEMDAS to different types of expressions—including those with multiple operations, nested parentheses, and exponents—the more natural and automatic it will become. Whether you're tackling homework problems, standardized test questions, or real-world calculations, PEMDAS gives you the reliable framework to navigate any expression with confidence.
Bookmark this guide, share it with friends who might be struggling, and use the cheat sheet whenever you need a quick refresher. With consistent practice and a solid understanding of these rules, you'll master the order of operations and never get tripped up by a complex expression again.
