Order of Operations (PEMDAS)

Why This Matters

Every algebraic expression you'll encounter—whether it's a simple calculation or a complex equation—depends on you knowing exactly which operation to perform first. PEMDAS isn't just a memory trick; it's the universal agreement mathematicians use so that $3 + 4 \times 2$ means the same thing to everyone (it's 11, not 14). You're being tested on your ability to evaluate expressions correctly, and a single misstep in operation order can derail an entire problem.

The good news? Once you understand why operations are prioritized the way they are—grouping symbols create boundaries, exponents are compact multiplication, and multiplication/division outrank addition/subtraction—the rules become intuitive. Don't just memorize "Please Excuse My Dear Aunt Sally." Know what each step accomplishes and how grouping symbols change everything.

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Grouping Symbols: Creating Boundaries

Grouping symbols tell you "solve this part first" by creating a protected zone. Think of them as mathematical fences—nothing outside the fence can touch what's inside until you've simplified it completely.

Parentheses

• Parentheses ( ) are your first priority—always evaluate what's inside before doing anything else
• Nested parentheses require working from the innermost set outward, like peeling layers of an onion
• Strategic placement of parentheses changes meaning entirely: $(3 + 4) \times 2 = 14$ but $3 + (4 \times 2) = 11$

Brackets and Braces

• Brackets [ ] and braces { } function identically to parentheses—they're just visual organizers for complex expressions
• Layered grouping typically follows the pattern $\{[( \text{innermost} )]\}$ to help you track which level you're solving
• Work inside-out, completing each grouping level before moving to the next outer layer

Fraction Bars as Grouping Symbols

• Fraction bars act as invisible parentheses—the entire numerator and entire denominator are separate groups
• Simplify top and bottom independently before performing the division the fraction bar represents
• Common error alert: In $\frac{6 + 2}{4}$, you must add first to get $\frac{8}{4} = 2$, not $\frac{6}{4} + 2$

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Powers: Compact Multiplication

Exponents represent repeated multiplication, which is why they're evaluated before regular multiplication—you need to know what the base actually equals before you can use it in other operations.

Exponents

• Exponents are evaluated second, immediately after resolving all grouping symbols
• The expression $2^3$ means $2 \times 2 \times 2 = 8$—this must be calculated before any multiplication with other terms
• Roots count as exponents because $\sqrt{x} = x^{1/2}$; evaluate them at this same stage

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Equal-Priority Operations: Left to Right

Here's where students make the most mistakes. Multiplication doesn't always come before division, and addition doesn't always come before subtraction. Pairs of operations at the same level are performed left to right, like reading a sentence.

Multiplication and Division

• Equal priority means left-to-right order—in $12 \div 3 \times 2$, divide first to get $4 \times 2 = 8$
• These are inverse operations, which is why they share the same priority level in the hierarchy
• Common trap: Seeing multiplication and jumping ahead; always scan left to right for both operations

Addition and Subtraction

• Also equal priority, also left to right—in $10 - 4 + 2$, subtract first to get $6 + 2 = 8$
• This is the final step after all grouping, exponents, and multiplication/division are complete
• Check your work by confirming no higher-priority operations remain before adding or subtracting

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Strategic Simplification

Smart mathematicians look for ways to make expressions easier before grinding through calculations. This isn't cheating—it's efficiency.

Simplify Before Solving

• Combine like terms early when possible—$3x + 2x$ becomes $5x$ before you do anything else with it
• Reduce fractions before multiplying to avoid large numbers: $\frac{4}{8} \times 16$ is easier as $\frac{1}{2} \times 16 = 8$
• Simplification prevents errors by keeping numbers manageable throughout your calculations

Use Parentheses to Clarify

• Add parentheses to ambiguous expressions to make your intended order crystal clear
• When writing your own expressions, parentheses communicate your thinking to graders and prevent misreading
• Readability matters—$2 \times (3 + 4)$ is clearer than relying on others to remember PEMDAS perfectly

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Quick Reference Table

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Self-Check Questions

1. In the expression $8 \div 2 \times 4$, which operation do you perform first, and why?

2. What do parentheses, brackets, braces, and fraction bars all have in common in terms of how they affect order of operations?

3. Compare and contrast how you handle multiplication/division versus addition/subtraction—what rule applies to both pairs?

4. A student evaluates $\frac{4 + 8}{2}$ as $4 + 4 = 8$. What error did they make, and what is the correct answer?

5. Given the expression $5 + 3 \times 2^2 - 4$, list the order in which you would perform each operation and state the final answer.