Order of Operations Rules

Why This Matters

The order of operations isn't just a random set of rules—it's the universal language that ensures everyone gets the same answer from the same expression. You're being tested on your ability to decode mathematical expressions correctly, which means understanding why certain operations take priority over others. Without these rules, an expression like $2 + 3 \times 4$ could mean 20 or 14 depending on who's solving it.

Mastering order of operations builds the foundation for everything else in algebra: solving equations, simplifying expressions, and working with functions. The key concepts here are grouping symbols, hierarchy of operations, and left-to-right processing. Don't just memorize PEMDAS as a word—know what each step does and why it comes in that order. That's what separates students who get tricked by test questions from those who breeze through them.

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Grouping Symbols: The Override Button

Grouping symbols like parentheses and brackets act as mathematical instructions that say "do this first, no matter what." They override the normal hierarchy because they explicitly mark priority.

Parentheses Come First

• Parentheses signal priority—any operation inside $(\ )$ must be completed before you touch anything outside
• Nested parentheses work inside-out; solve the innermost grouping first, then work your way outward
• Strategic placement of parentheses can completely change an answer: $(2 + 3) \times 4 = 20$ vs. $2 + (3 \times 4) = 14$

Brackets and Other Grouping Symbols

• Brackets $[\ ]$ function identically to parentheses—they're often used as outer groupings to reduce visual confusion
• Fraction bars act as invisible parentheses—the entire numerator and entire denominator are each treated as grouped expressions
• Absolute value bars and radical symbols also create implicit groupings that must be resolved first

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The Power Tier: Exponents

Exponents represent repeated multiplication, which makes them more "powerful" than single multiplication operations. They compact multiple operations into one notation, so they must be evaluated before multiplication can proceed.

Evaluate Exponents Second

• Exponents apply only to what they're directly attached to—in $-3^2$, only the 3 is squared, giving $-9$, not $9$
• Parentheses change exponent scope—$(-3)^2 = 9$ because the negative is inside the grouping
• Calculate all exponents before moving to multiplication/division, even if the exponent appears later in the expression

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Equal Priority Pairs: Left to Right Processing

Multiplication/division and addition/subtraction each form pairs of equal priority. Within each pair, operations are processed left to right—not multiplication before division or addition before subtraction.

Multiplication and Division

• Equal priority, left to right—in $24 \div 4 \times 2$, divide first to get $6 \times 2 = 12$, not $24 \div 8 = 3$
• Inverse operations—multiplication and division undo each other, which is why they share the same level
• Don't be fooled by PEMDAS—the "M before D" in the acronym is just alphabetical, not mathematical priority

Addition and Subtraction

• Equal priority, left to right—process $10 - 3 + 2$ as $(10 - 3) + 2 = 9$, not $10 - (3 + 2) = 5$
• Subtraction is adding a negative—thinking of $10 - 3$ as $10 + (-3)$ can help avoid sign errors
• Final step in simplification—addition and subtraction close out every order of operations problem

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Working with Fractions

Fractions introduce special considerations because they contain implicit grouping and can be simplified independently. The fraction bar acts as both a division symbol and a grouping symbol simultaneously.

Treat Numerator and Denominator Separately

• Each part follows full PEMDAS—simplify $\frac{3 + 5}{2^2}$ by resolving the numerator $(8)$ and denominator $(4)$ independently, then divide
• Simplify fractions when possible—reducing before other operations prevents unwieldy numbers
• The fraction bar means division—$\frac{a + b}{c}$ is equivalent to $(a + b) \div c$, not $a + b \div c$

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Quick Reference Table

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Self-Check Questions

1. What is the value of $8 - 2 \times 3 + 4$? Which operations did you perform first, and why?

2. Compare $-5^2$ and $(-5)^2$. Why do they give different answers, and which concept explains the difference?

3. In the expression $\frac{4 + 8}{3 + 1}$, what role does the fraction bar play in determining order of operations?

4. A student solves $12 \div 2 \times 3$ and gets $2$. What error did they make, and what's the correct answer?

5. Write an expression using only the numbers 2, 3, and 4 (each used once) and any operations/parentheses that equals 14. Then write another that equals 10. What does this demonstrate about parentheses?