Order of Operations Rules

Why This Matters

The order of operations is the universal agreement that ensures everyone gets the same answer from the same expression. 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 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 tell you "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.
• 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 $[\ ]$ work the same as parentheses. They're often used as outer groupings to reduce visual confusion when you already have parentheses inside.
• 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 groupings that must be resolved first.

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The Power Tier: Exponents

Exponents represent repeated multiplication, which makes them more "powerful" than a single multiplication. 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 or division, even if the exponent appears later in the expression.

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Equal Priority Pairs: Left to Right Processing

Here's where PEMDAS misleads people. Multiplication and division share the same priority. Addition and subtraction share the same priority. Within each pair, you process left to right. It's not "multiplication before division" or "addition before subtraction."

Multiplication and Division

• Equal priority, left to right. In $24 \div 4 \times 2$, you divide first because $\div$ comes before $\times$ reading left to right: $6 \times 2 = 12$, not $24 \div 8 = 3$.
• They're inverse operations. Multiplication and division undo each other, which is why they share the same level.
• The "M before D" in PEMDAS is just alphabetical order, not mathematical priority. A more accurate way to think of it is P-E-MD-AS, where MD and AS are tied pairs.

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 you avoid sign errors.
• This is always your final step. Addition and subtraction close out every order of operations problem.

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Working with Fractions

Fractions need special attention because they contain implicit grouping. The fraction bar acts as both a division symbol and a grouping symbol at the same time.

Treat Numerator and Denominator Separately

1. Simplify the numerator using the full order of operations.
2. Simplify the denominator using the full order of operations.
3. Divide the simplified numerator by the simplified denominator.

For example, with $\frac{3 + 5}{2^2}$: the numerator simplifies to $8$, the denominator simplifies to $4$, and then $8 \div 4 = 2$.

• 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?