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Using PEMDAS to evaluate expressions
Learn step-by-step with practice exercises built right in.
When an expression has multiple operations, which do you do first? The order of operations ensures everyone gets the same answer!
Order of operations is a set of rules that tells you the correct sequence for evaluating mathematical expressions.
Without rules: 5 + 3 × 2 = ?
Some might think: 5 + 3 = 8, then 8 × 2 = 16 Others might think: 3 × 2 = 6, then 5 + 6 = 11
Which is right? We need a standard order!
PEMDAS is an acronym to remember the order:
P - Parentheses (and other grouping symbols) E - Exponents (powers and roots) M - Multiplication D - Division A - Addition S - Subtraction
Memory trick: "Please Excuse My Dear Aunt Sally"
Evaluate:
Follow PEMDAS: Multiplication before Addition
Step 1: Multiply first
Avoid these 3 frequent errors
See how this math is used in the real world
Solve .
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Key point: M and D have the SAME priority! Do them left to right as they appear.
Same with A and S - equal priority, left to right.
Better to think of PEMDAS as:
Do operations inside parentheses FIRST.
Example: 3 + (4 × 2)
Step 1: Parentheses → 4 × 2 = 8 Step 2: Addition → 3 + 8 = 11
Answer: 11
Example 2: (5 + 3) × 2
Step 1: Parentheses → 5 + 3 = 8 Step 2: Multiplication → 8 × 2 = 16
Answer: 16
Without parentheses: 5 + 3 × 2 = 5 + 6 = 11 (different!)
After parentheses, do exponents.
Example: 2 + 3²
Step 1: Exponents → 3² = 9 Step 2: Addition → 2 + 9 = 11
Answer: 11
Example 2: 5 × 2³ - 10
Step 1: Exponents → 2³ = 8 Step 2: Multiplication → 5 × 8 = 40 Step 3: Subtraction → 40 - 10 = 30
Answer: 30
After exponents, do multiplication and division AS THEY APPEAR from left to right.
Example: 12 ÷ 3 × 2
Left to right: 12 ÷ 3 = 4 4 × 2 = 8
Answer: 8
NOT 12 ÷ (3 × 2) = 12 ÷ 6 = 2 ❌
Example 2: 20 × 2 ÷ 5
Left to right: 20 × 2 = 40 40 ÷ 5 = 8
Answer: 8
Last, do addition and subtraction AS THEY APPEAR from left to right.
Example: 10 - 3 + 2
Left to right: 10 - 3 = 7 7 + 2 = 9
Answer: 9
NOT 10 - (3 + 2) = 10 - 5 = 5 ❌
Example 2: 5 + 8 - 4 + 2
Left to right: 5 + 8 = 13 13 - 4 = 9 9 + 2 = 11
Answer: 11
Evaluate: 3 + 4 × 2² - 5
Step 1: Parentheses (none)
Step 2: Exponents 2² = 4 Expression: 3 + 4 × 4 - 5
Step 3: Multiplication/Division 4 × 4 = 16 Expression: 3 + 16 - 5
Step 4: Addition/Subtraction (left to right) 3 + 16 = 19 19 - 5 = 14
Answer: 14
Work from inside out!
Example: 2 × (3 + (4 × 5))
Step 1: Innermost parentheses 4 × 5 = 20 Expression: 2 × (3 + 20)
Step 2: Next parentheses 3 + 20 = 23 Expression: 2 × 23
Step 3: Multiply 2 × 23 = 46
Answer: 46
Different grouping symbols, same priority as parentheses:
( ) - Parentheses [ ] - Brackets { } - Braces
All mean "do this first!"
Example: 5 + [3 × (2 + 1)]
Inside to out: 2 + 1 = 3 3 × 3 = 9 5 + 9 = 14
Answer: 14
The fraction bar groups the numerator and denominator.
Example: (6 + 4) / (5 - 3)
Think of it as: (6 + 4) ÷ (5 - 3)
Numerator: 6 + 4 = 10 Denominator: 5 - 3 = 2 Division: 10 ÷ 2 = 5
Answer: 5
Many students incorrectly do this: 5 + 3 = 8 8 × 2 = 16 ❌ WRONG!
Correct order: Multiplication first: 3 × 2 = 6 Then addition: 5 + 6 = 11 ✓
Remember: Multiplication before addition!
Evaluate: 20 ÷ 4 + 3² × 2 - 1
Step 1: Exponents 3² = 9 Expression: 20 ÷ 4 + 9 × 2 - 1
Step 2: Multiplication/Division (left to right) 20 ÷ 4 = 5 9 × 2 = 18 Expression: 5 + 18 - 1
Step 3: Addition/Subtraction (left to right) 5 + 18 = 23 23 - 1 = 22
Answer: 22
Example: 8 ÷ 4 × 2
Correct (left to right): 8 ÷ 4 = 2 2 × 2 = 4 ✓
Incorrect (if you did multiplication first): 4 × 2 = 8 8 ÷ 8 = 1 ❌ WRONG!
Always left to right for operations of equal priority!
Example: -3² vs (-3)²
-3²: Exponent first, then negative 3² = 9 Then apply negative: -9
(-3)²: Parentheses include the negative (-3) × (-3) = 9
Very different answers!
Example 2: 5 + -3
This is really 5 + (-3) = 2
Example: (8 - 3) × (2 + 4)
Step 1: Do both parentheses 8 - 3 = 5 2 + 4 = 6
Step 2: Multiply results 5 × 6 = 30
Answer: 30
Example: 2x + 3² when x = 4
Step 1: Substitute 2(4) + 3²
Step 2: Exponents 2(4) + 9
Step 3: Multiplication 8 + 9
Step 4: Addition 17
Answer: 17
Shopping: You buy 3 shirts at 20 hat. Total cost?
Expression: 3 × 15 + 20
Correct: Multiply first: 3 × 15 = 45 Add: 45 + 20 = 65
Total: $65
If you added first (wrong): 15 + 20 = 35 3 × 35 = 105 ❌ WRONG!
Rectangle: length 5, width (3 + 2)
Perimeter: 2(5 + 3 + 2) Parentheses: 5 + 3 + 2 = 10 Multiply: 2 × 10 = 20
Or: 2 × 5 + 2(3 + 2) = 10 + 2(5) = 10 + 10 = 20
Same answer both ways!
Square root acts like parentheses for what's underneath.
Example: √(16 + 9)
Add first: 16 + 9 = 25 Then square root: √25 = 5
Answer: 5
NOT √16 + √9 = 4 + 3 = 7 ❌
Evaluate: 100 - 4(2³ - 5) + 10 ÷ 2
Step 1: Parentheses (inside first) Exponent in parentheses: 2³ = 8 Subtraction: 8 - 5 = 3 Expression: 100 - 4(3) + 10 ÷ 2
Step 2: Multiplication/Division (left to right) 4(3) = 12 10 ÷ 2 = 5 Expression: 100 - 12 + 5
Step 3: Addition/Subtraction (left to right) 100 - 12 = 88 88 + 5 = 93
Answer: 93
❌ Mistake 1: Adding before multiplying
❌ Mistake 2: Not going left to right
❌ Mistake 3: Forgetting parentheses change everything
❌ Mistake 4: Squaring negative incorrectly
❌ Mistake 5: Not working inside out with nested parentheses
PEMDAS: Please Excuse My Dear Aunt Sally
Alternative: GEMDAS
Alternative: BODMAS (British)
All mean the same thing!
To evaluate any expression:
Scan for parentheses (or brackets/braces)
Look for exponents (and roots)
Find multiplication and division
Do addition and subtraction
Check your work
Most calculators follow order of operations automatically.
Try: Enter 5 + 3 × 2
Scientific calculator: 11 ✓ Basic calculator might give: 16 ❌ (depends on calculator)
When in doubt, use parentheses to be clear!
Ambiguous: 6 ÷ 2 × 3
Could be interpreted differently. Better to write:
(6 ÷ 2) × 3 = 9 or 6 ÷ (2 × 3) = 1
Use parentheses to make your intent clear!
All × before +:
Parentheses change everything:
Exponents before multiplication:
Order:
Remember:
PEMDAS = Please Excuse My Dear Aunt Sally
To change the order:
To make clear what you mean:
In algebra:
Order of operations (PEMDAS) ensures everyone gets the same answer:
Priority from highest to lowest:
Key principles:
Applications:
Common errors to avoid:
Master PEMDAS and you'll evaluate any expression correctly every time!
Step 2: Add
Answer:
Evaluate: 8 + 3 × 4
Step 1: Follow PEMDAS order. P - Parentheses (none) E - Exponents (none) M/D - Multiplication/Division (left to right) A/S - Addition/Subtraction (left to right)
Step 2: Do multiplication FIRST. 3 × 4 = 12
Step 3: Then do addition. 8 + 12 = 20
Common mistake: 8 + 3 = 11, then 11 × 4 = 44 ✗
Answer: 20
Evaluate:
Follow PEMDAS: Multiplication before Addition
Step 1: Multiply first
Step 2: Add
Answer:
Evaluate:
Step 1: Division and multiplication (left to right)
Step 2: Addition
Answer:
Evaluate:
Step 1: Division and multiplication (left to right)
Step 2: Addition
Answer:
Calculate: (6 + 2) × 5
Step 1: PEMDAS - Parentheses first! (6 + 2) = 8
Step 2: Now multiply. 8 × 5 = 40
Answer: 40
Evaluate:
Step 1: Innermost parentheses
Step 2: Brackets
Step 3: Multiply
Step 4: Add
Answer:
Evaluate:
Step 1: Innermost parentheses
Step 2: Brackets
Step 3: Multiply
Step 4: Add
Answer:
Evaluate: 20 - 12 ÷ 3 + 4
Step 1: Follow PEMDAS. No parentheses or exponents. Division comes before addition/subtraction.
Step 2: Do division first. 12 ÷ 3 = 4
Rewrite: 20 - 4 + 4
Step 3: Add/subtract from left to right. 20 - 4 = 16 16 + 4 = 20
Answer: 20
Calculate: 3² + 4(5 - 2)
Step 1: Parentheses first. (5 - 2) = 3
Rewrite: 3² + 4(3)
Step 2: Exponents. 3² = 9
Rewrite: 9 + 4(3)
Step 3: Multiplication. 4 × 3 = 12
Rewrite: 9 + 12
Step 4: Addition. 9 + 12 = 21
Answer: 21
Evaluate: 2[(8 - 3) × 2 + 6] - 4²
Step 1: Start with innermost parentheses. (8 - 3) = 5
Rewrite: 2[5 × 2 + 6] - 4²
Step 2: Operations inside brackets (multiply first). 5 × 2 = 10
Rewrite: 2[10 + 6] - 4²
Step 3: Continue inside brackets (add). 10 + 6 = 16
Rewrite: 2[16] - 4²
Step 4: Exponents. 4² = 16
Rewrite: 2(16) - 16
Step 5: Multiplication. 2 × 16 = 32
Rewrite: 32 - 16
Step 6: Subtraction. 32 - 16 = 16
Answer: 16