0.2Order of Operations
Learn the rules of PEMDAS — Parentheses, Exponents, Multiplication/Division, Addition/Subtraction — so every expression has exactly one correct answer.
Why This Matters
Order of operations is the universal grammar of math — without it, the same expression could give different answers. You'll apply PEMDAS in every future math and science course, from evaluating physics formulas to programming calculations.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
Why do mathematicians need an agreed-upon order of operations, and what happens when we ignore it?
Lesson Overview
Without a standard order of operations, the expression 2 + 3 × 4 could equal 20 or 14 depending on how you read it. Mathematicians agreed on a universal set of rules called the order of operations (remembered by the acronym PEMDAS) so that every expression has exactly one correct value. The rules are: (1) Parentheses/grouping symbols, (2) Exponents, (3) Multiplication and Division left to right, (4) Addition and Subtraction left to right.
Worked Examples
Simplify: 2 + 3 × 4
No parentheses or exponents.
Multiply first: 3 × 4 = 12
Add: 2 + 12 = 14
Simplify: (5 + 3)² − 6 ÷ 2
Parentheses: 5 + 3 = 8
Exponent: 8² = 64
Division: 6 ÷ 2 = 3
Subtract: 64 − 3 = 61
Simplify: 3 × [4 + (2² − 1)]
Inner parentheses — exponent: 2² = 4
Inner parentheses — subtract: 4 − 1 = 3
Brackets: 4 + 3 = 7
Multiply: 3 × 7 = 21
Simplify: 18 ÷ 3 + 4 × 2 − 5
Divide: 18 ÷ 3 = 6
Multiply: 4 × 2 = 8
Left to right: 6 + 8 − 5 = 9
Simplify: 5 + 2(3² − 4)
Parentheses — exponent: 3² = 9
Parentheses — subtract: 9 − 4 = 5
Multiply: 2 × 5 = 10
Add: 5 + 10 = 15
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Order of Operations
Watch the guided practice walkthrough for order of operations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Simplify: 4 + 6 ÷ 2
Hint: Division comes before addition. Divide first.
Simplify: (7 − 3) × 5 + 2
Hint: Work inside the parentheses first, then multiply, then add.
Simplify: 2³ + 4 × 3 − 1
Hint: Exponent first, then multiply, then add/subtract left to right.
Simplify: 10 − 2 × (3 + 1)
Hint: Parentheses first, then multiply, then subtract.
Simplify: [6 + 2(4 − 1)] ÷ 3
Hint: Start with the innermost grouping: (4 − 1). Then work outward.
Key Vocabulary
PEMDAS
Acronym for the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Example: In 3 + 2 × 4, multiply first: 3 + 8 = 11.
Grouping Symbols
Parentheses ( ), brackets [ ], and braces { } that indicate which operations to perform first.
Example: (3 + 2) × 4 = 5 × 4 = 20
Exponent
A number that tells how many times the base is multiplied by itself.
Example: 3² = 3 × 3 = 9
Order of Operations
The agreed-upon rules for the sequence in which operations must be performed.
Example: Always: P → E → MD → AS
Expression
A mathematical phrase with numbers and/or variables combined with operations — no equals sign.
Example: 4 + 3 × 2 or 5x − 1
Simplify
To perform all operations and write the expression in its simplest form.
Example: Simplify 2 + 3 × 4 → 14
Practice Questions
Interactive Practice — 5 Questions
Simplify: 2 + 3 × 4
Simplify: (5 + 3)² − 6 ÷ 2
Simplify: 18 ÷ 3 + 4 × 2 − 5
Simplify: 5 + 2(3² − 4)
Simplify: 3 × [4 + (2² − 1)]
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Simplify: 3 + 5 × 2
Simplify: (3 + 5) × 2
Simplify: 4² − 3 × 2 + 1
Simplify: 2 × (6 − 4)² + 3
Simplify: [10 − 2 × (1 + 3)] ÷ 2
Common Mistakes
Adding or subtracting before multiplying or dividing — e.g., computing 2 + 3 × 4 as 5 × 4 = 20.
Multiplication and division come before addition and subtraction. 2 + 3 × 4 = 2 + 12 = 14.
Forgetting that exponents are evaluated before multiplication — e.g., treating 2 × 3² as (2 × 3)² = 36.
Evaluate the exponent first: 2 × 3² = 2 × 9 = 18.
Skipping the brackets/parentheses and working left to right through everything.
Always resolve the innermost grouping symbols first, then work outward.
Treating multiplication and division as if multiplication always comes first.
Multiplication and division have equal priority — work them left to right as they appear.
Math Tips
Use the phrase "Please Excuse My Dear Aunt Sally" to remember PEMDAS — but remember M/D and A/S are equal pairs, worked left to right.
When simplifying multi-step expressions, rewrite the expression after each step. This prevents errors and makes it easy to check your work.
On a calculator, always use parentheses to group operations the way you intend. Calculators follow PEMDAS, but misplaced entries cause wrong answers.
SAT/ACT tip: order-of-operations errors are one of the most common mistakes on standardized tests. When in doubt, slow down and work one step at a time.