Grouping symbols, exponents, multiplication and division, addition and subtraction — the shared agreement that makes mathematical communication reliable
Mathematics is a language. Like any language, it needs rules so that the people using it understand each other. When you write a sentence in English, word order changes meaning. "The dog bit the man" says something very different from "The man bit the dog," even though both sentences use the same words.
Mathematical expressions work the same way. When an expression contains more than one operation, the order in which those operations are performed changes the result. Without shared rules, two students could evaluate the same expression and reach two completely different answers, and both could believe they were correct.
Order of operations is not an arbitrary list of rules handed down for no reason. It is an agreement, a shared convention that makes mathematical communication reliable. This chapter teaches that agreement carefully. It begins by showing why the rules are necessary, then builds through grouping symbols, exponents, multiplication and division, addition and subtraction, and increasingly complex expressions. You will also learn to evaluate expressions involving whole numbers, negative numbers, fractions, decimals, nested parentheses, and formulas.
Two students evaluate the same expression: 3 + 4 × 2. Student A adds first, then multiplies: 3 + 4 = 7, then 7 × 2 = 14. Student B multiplies first, then adds: 4 × 2 = 8, then 3 + 8 = 11. Only one student is correct. Which one, and why? By the end of this chapter, you will not only know the answer — you will understand the principle behind it.
Order of operations appears in every topic that follows this chapter. When you evaluate algebraic expressions, simplify formulas, use a calculator, write a spreadsheet formula, solve physics problems, or read any mathematical text, you are relying on the order of operations every time. Students who treat this chapter as minor often make consistent errors in Algebra I, geometry, and chemistry, not because they do not know arithmetic, but because they evaluate expressions in the wrong order.
−3² from (−3)².Consider the expression 2 + 3 × 4. This expression contains two operations: addition and multiplication. If you work left to right, you get 2 + 3 = 5, then 5 × 4 = 20. If you multiply before you add, you get 3 × 4 = 12, then 2 + 12 = 14. Both computations involve correct arithmetic. The question is which one represents what the expression actually means. Without a rule, the expression is ambiguous — it could mean two different things to two different people.
Mathematics depends on precise communication. If an engineer, scientist, accountant, or teacher writes an expression, every reader must interpret it identically. The agreed-upon rule, developed and standardized over centuries of mathematical practice, is that multiplication is performed before addition unless grouping symbols indicate otherwise. Under this agreement, 2 + 3 × 4 = 14, not 20.
Grouping symbols override this priority. If you want addition to happen first, you can force it with parentheses: (2 + 3) × 4 = 20. Parentheses give you the power to say exactly what you mean.
1st — Evaluate anything inside grouping symbols, innermost group first.
2nd — Evaluate exponents.
3rd — Perform multiplication and division from left to right.
4th — Perform addition and subtraction from left to right.
PEMDAS is a memory tool, not a strict six-step checklist. Multiplication and division are at the same priority level. Addition and subtraction are at the same priority level. The left-to-right rule governs what happens when operations at the same level appear together.
10 − 2 × 3 using correct order. Then show what happens working left to right.10 − 2 × 3 = 10 − 6 = 410 − 2 = 8, then 8 × 3 = 24 ← incorrectConcept Check: 1–3 | Skill Practice: 4–7 | Application: 8
5 + 2 × 6 does not equal 42.8 + 3 × 212 − 4 × 25 × 3 + 420 − 3 × 5 + 14 + 6 × 2 to represent a total cost and evaluates it as 20. Is that correct? What should the answer be?Grouping symbols are tools that tell you to evaluate a portion of an expression before applying the surrounding operations. They are the highest priority step in the order of operations. No matter what other operations appear in an expression, anything inside grouping symbols must be simplified first. The most common grouping symbols are parentheses ( ), brackets [ ], and braces { }. The fraction bar, the radical sign, and absolute value bars also function as grouping symbols.
Parentheses are the most familiar. When you see an expression inside parentheses, evaluate that expression completely before moving on. Compare (3 + 5) × 2 = 8 × 2 = 16 to the same numbers without parentheses: 3 + 5 × 2 = 3 + 10 = 13. The parentheses change the answer entirely. They let the writer override the default order and say, evaluate this part first.
The fraction bar groups both the numerator and denominator separately. Both must be fully simplified before the division is performed. Absolute value bars group the expression inside them; the expression is evaluated first, then the absolute value is applied.
Some students find the absolute value of each number separately before subtracting. |10| − |14| = 10 − 14 = −4 is not the same as |10 − 14| = 4. The subtraction happens inside the bars first.
(6 + 2) × (5 − 3).(6 + 2) × (5 − 3) = 8 × 2 = 16(5 + 7) ÷ (3 × 2), written as a fraction.Numerator: 5 + 7 = 12
Denominator: 3 × 2 = 6
Result: 12 ÷ 6 = 22 × |10 − 14|.2 × |10 − 14| = 2 × |−4| = 2 × 4 = 8(4 + 3) × 2 = 4 + 3 × 2.(8 − 3) × 46 + (2 × 5)(10 + 6) ÷ 4(9 − 3) ÷ (2 × 1)3 × |7 − 10||12 − 4| + |3 − 9|12 ÷ (2 + 4) as 6 + 4 = 10. Identify and correct the error.After grouping symbols have been evaluated, exponents are next. When you encounter an expression with both an exponent and another operation, evaluate the exponent before performing multiplication, division, addition, or subtraction, unless a grouping symbol tells you otherwise.
A critical point: an exponent applies only to its base, which is the number or group immediately to its left. Pay careful attention to what is and what is not inside the base.
−3² means the negative of 3². The exponent applies only to 3, not the negative sign.
−3² = −(3²) = −9
(−3)² means the quantity (−3) is being squared. The parentheses make −3 the base.
(−3)² = (−3) × (−3) = +9
Squaring a sum is not the same as summing the squares. (3 + 4)² = 7² = 49, which is not equal to 3² + 4² = 9 + 16 = 25. Always evaluate the group inside parentheses first, then apply the exponent. This error is extremely common and becomes very important in algebra.
4 + 3² × 2.4 + 3² × 2
= 4 + 9 × 2
= 4 + 18
= 22−5² and (−5)². Explain the difference.−5² = −(5 × 5) = −25 [exponent applies only to 5]
(−5)² = (−5)(−5) = +25 [parentheses make −5 the base](1 + 4)² − 3.(1 + 4)² − 3
= 5² − 3
= 25 − 3
= 222 + 4², which operation is performed first?−6² and (−6)²?(3 + 4)² ≠ 3² + 4².5 + 2⁴(5 + 2)⁴10 − 3²−4²(−4)²2³ + 4² − 10(2 + 1)³ − (1 + 1)²Multiplication and division have the same priority in the order of operations. They are not in a strict sequence where all multiplications happen before all divisions. Instead, when multiplication and division appear together in an expression, you perform them from left to right in the order you encounter them.
Because multiplication comes before division in the acronym PEMDAS, many students think all multiplication must be completed before any division. This is incorrect. They are equal in priority. Work left to right. For 12 ÷ 4 × 3: divide first (it's on the left): 12 ÷ 4 = 3, then multiply: 3 × 3 = 9. Not 4 × 3 = 12, then 12 ÷ 12 = 1.
20 ÷ 5 × 2.20 ÷ 5 × 2 = 4 × 2 = 85 + 6 × 3 ÷ 9 − 1.Step 1 — M/D left to right: 6 × 3 = 18, then 18 ÷ 9 = 2
Step 2 — A/S: 5 + 2 − 1 = 624 ÷ 6 × 23 × 8 ÷ 418 ÷ 3 ÷ 27 + 4 × 3 − 215 − 2 × 4 + 12 + 30 ÷ 5 × 3 − 420 ÷ 4 × 5 by computing 4 × 5 = 20, then 20 ÷ 20 = 1. What rule was violated?100 ÷ 10 ÷ 5 × 3Addition and subtraction share the lowest priority in the standard order of operations. They are evaluated after all grouping symbols, exponents, and multiplication and division have been handled. Like multiplication and division, they share a level: when they appear together, work from left to right.
Consider 10 − 3 + 2. Working left to right: 10 − 3 = 7, then 7 + 2 = 9. If you added first: 3 + 2 = 5, then 10 − 5 = 5. That would be wrong. The left-to-right rule governs the result.
Some students think all addition must be done before all subtraction because A appears before S in PEMDAS. This is a misreading of the acronym. Addition and subtraction are at the same level. Work left to right.
| Priority | Operation(s) | Rule |
|---|---|---|
| 1st | Grouping symbols | Evaluate from innermost outward |
| 2nd | Exponents | Apply to the base only |
| 3rd | Multiplication and Division | Left to right — equal priority |
| 4th | Addition and Subtraction | Left to right — equal priority |
15 − 8 + 3 − 2.15 − 8 = 7, 7 + 3 = 10, 10 − 2 = 83² + 4 × 2 − (5 + 1).Step 1 — Grouping: (5 + 1) = 6
Step 2 — Exponent: 3² = 9
Step 3 — Multiply: 4 × 2 = 8
Step 4 — A/S: 9 + 8 − 6 = 1118 − 6 + 420 − 3 − 5 + 22² + 3 × 4 − 610 − (2 + 3) + 4²5 × 2 + 3³ − (4 × 5)(6 − 2)² − 3 × 4 + 1Now that each component of PEMDAS has been introduced separately, this section applies all the steps together to whole number expressions. A reliable process for evaluating any expression is: read the entire expression before writing anything; identify grouping symbols and evaluate the innermost first; evaluate exponents; evaluate multiplication and division from left to right; evaluate addition and subtraction from left to right; write the final result. Show your work step by step, rewriting the expression after each step.
Write the expression again after each step, replacing only what you just computed. This keeps the rest of the expression visible and reduces copying errors. Each rewrite is a checkpoint.
2 + 3² × (8 − 5) − 4.Step 1 — Parentheses: (8 − 5) = 3
→ 2 + 3² × 3 − 4
Step 2 — Exponent: 3² = 9
→ 2 + 9 × 3 − 4
Step 3 — Multiply: 9 × 3 = 27
→ 2 + 27 − 4
Step 4 — A/S: 2 + 27 = 29; 29 − 4 = 2540 ÷ (2³) + 5 × 3 − 6.Step 1 — Exponent inside ( ): 2³ = 8
→ 40 ÷ 8 + 5 × 3 − 6
Step 2 — M/D left to right: 40 ÷ 8 = 5; 5 × 3 = 15
→ 5 + 15 − 6
Step 3 — A/S: 5 + 15 = 20; 20 − 6 = 145 + 2 × 3²(5 + 2) × 3²18 ÷ (2 + 1) + 4²7 × 2 − 3² + 10 ÷ 5(3 + 1)² − 2 × 5 + 3100 ÷ 5² + (3 × 4) − 6Negative numbers interact with the order of operations in ways that students often find confusing, particularly around the placement of negative signs, signed multiplication and division, and the distinction between subtraction and negation. A negative sign in front of a group, such as −(4 + 3), means you evaluate the group first and then negate the result: −(4 + 3) = −7. This is different from −4 + 3 = −1, where the negative sign applies only to 4. When a negative number appears in an expression with other operations, apply the multiplication or division first, keeping track of the sign.
(−2)³ + 4 × (−3) − 5.Step 1 — Exponent: (−2)³ = (−2)(−2)(−2) = −8
→ −8 + 4 × (−3) − 5
Step 2 — Multiply: 4 × (−3) = −12
→ −8 + (−12) − 5
Step 3 — A/S: −8 + (−12) = −20; −20 − 5 = −25−2⁴ + (−2)⁴.−2⁴ = −(2⁴) = −16 [exponent on 2 only]
(−2)⁴ = (−2)(−2)(−2)(−2) = +16
−16 + 16 = 0(−3)² − 4 × 2−3² − 4 × 26 + (−4) × 3 − 1−(2 + 5)² + 10(−1)⁵ × 8 − 3(−2)³ + (−2)² + (−2)10 − (−2)² + (−2)³−4² as 16. Identify the error. What is the correct answer?Fractions require two additional considerations. First, the fraction bar is a grouping symbol, so simplify the numerator and denominator separately before dividing. Second, fractions within larger expressions participate in the multiplication-division level and must be evaluated in the correct left-to-right order.
(3² + 7) ÷ (4 − 2) + 5, written with a fraction bar.Numerator: 3² + 7 = 9 + 7 = 16
Denominator: 4 − 2 = 2
Division: 16 ÷ 2 = 8
Add: 8 + 5 = 134 + (3/4) × 8 − 2.(3/4) × 8 = 24/4 = 6
4 + 6 − 2 = 8(1/2) ÷ (1/4) + 3.(1/2) ÷ (1/4) = (1/2) × 4 = 2
2 + 3 = 5(10 + 6) ÷ 4 + 32 + (1/3) × 9 − 1(5² − 1) ÷ 6 × 3(4 + 8) ÷ (3 + 1) − 1/2(2/3) ÷ (1/6) + (5 − 3)²(3 + 9) ÷ (2 + 4) by computing 3 + 9/2 + 4. Explain the error.The order of operations does not change when decimals appear in an expression. Every rule from the previous sections applies exactly as before. The only additional care required is accurate decimal arithmetic: aligning decimal points when adding or subtracting, and tracking decimal places when multiplying. When a decimal expression is complex, it is helpful to estimate first. After evaluating, check whether your answer is close to the estimate. This catches errors in decimal placement.
3.5 + 2.4 × 5.2.4 × 5 = 12.0
3.5 + 12.0 = 15.5(1.2 + 0.8)² − 1.5 × 2.Step 1 — Parentheses: 1.2 + 0.8 = 2.0
Step 2 — Exponent: 2.0² = 4.0
Step 3 — Multiply: 1.5 × 2 = 3.0
Step 4 — Subtract: 4.0 − 3.0 = 1.00.5 × 4 + 6.0 ÷ 1.5 − 0.2.M/D left to right: 0.5 × 4 = 2.0; 6.0 ÷ 1.5 = 4.0
A/S: 2.0 + 4.0 − 0.2 = 5.86.4 + 0.5 × 8(2.5 − 0.5)³4.8 ÷ 1.6 + 3.2 × 2(1.5 + 2.5) × 3² − 10.0(3.6 + 0.4) ÷ (0.5 × 4) + 1.50.25 × 4 + 1.5 cups of flour. How many cups is that?$5.50 + 3 × $0.75 − $1.00 after adjustments. What is the final cost?Nested parentheses occur when grouping symbols appear inside other grouping symbols. The rule is straightforward: begin with the innermost group and work outward, one layer at a time. Mathematicians often use different grouping symbols at different nesting levels to make an expression easier to read. Parentheses are used at the innermost level, brackets at the next, and braces at the outermost level. All three work identically in arithmetic, but alternating them helps readers identify the layers. Always find the innermost group, the one with no further grouping symbols inside it, and evaluate that first. When you finish evaluating an inner group, rewrite the whole expression before continuing.
4 + {3 × [2 + (6 − 4)] − 5}.Innermost: (6 − 4) = 2
→ 4 + {3 × [2 + 2] − 5}
Next layer: [2 + 2] = 4
→ 4 + {3 × 4 − 5}
Inside { }: 3 × 4 = 12; 12 − 5 = 7
→ 4 + 7
Final: 4 + 7 = 11[(3 + 1)² − 4] × 2.Innermost: (3 + 1) = 4
Exponent: 4² = 16
Inside [ ]: 16 − 4 = 12
Multiply: 12 × 2 = 242 × [3 + (4 − 1)]{5 − [2 + (1 + 1)]} × 4[10 − (2² + 1)] + 63 + {[(4 + 2) × 2] − 5}{2 × [(5 − 3)³ + 1]} − 10An expression tree is a diagram that represents the structure of a mathematical expression. The tree shows which operations depend on which values, and in what order they should be evaluated. Expression trees make the hierarchy of PEMDAS visible as a structure rather than a sequence of rules. They are common in computer science, where programming languages and calculators use them to parse and evaluate mathematical input.
In an expression tree, the root represents the last operation to be performed, the one with the lowest priority. Moving downward, each node represents an earlier operation. The leaves at the very bottom represent individual numbers. To evaluate, you start at the leaves and work upward.
You do not need to draw a full tree every time. But when you make a mistake you cannot find, sketching the tree can help you see which operation should be evaluated last, and trace backward to find where the evaluation went wrong.
2 + [(6 − 2)² ÷ 4] × 3.Step 1: (6 − 2) = 4 → 2 + [4² ÷ 4] × 3
Step 2: 4² = 16 → 2 + [16 ÷ 4] × 3
Step 3: 16 ÷ 4 = 4 → 2 + 4 × 3
Step 4: 4 × 3 = 12 → 2 + 12
Step 5: 2 + 12 = 145 + 3 × 2. Label the root and lower nodes. Then evaluate.(5 + 3) × 2. How does the tree differ from problem 1?3 × [(2 + 4)² − 6] + 1{12 ÷ [(5 − 3) × 2]} + 4²Most errors in the order of operations come not from carelessness, but from a handful of persistent misunderstandings. This section names each one directly, explains why it is wrong, and shows a correction.
Correct: 12 ÷ 3 × 2 = 4 × 2 = 8 [division is on the left; go left to right]
Correct: 10 − 3 + 2 = 7 + 2 = 9 [work left to right at the same level]
Correct: −3² = −(3²) = −9 [exponent applies only to 3, not the negative sign]
Correct: (3 + 4)² = 7² = 49 [evaluate the group first, then apply the exponent]
Correct: (6 + 4) ÷ 2 = 10 ÷ 2 = 5 [the numerator is fully grouped]
Correct: Always find the innermost group first — the one with no further grouping inside — and work outward.
Fix: After evaluating, look back at the original expression. A mistake in an early step affects every step after it. The check habit is especially valuable in PEMDAS work.
5 + 10 ÷ 2 × 5 − 3 by multiplying 2 × 5 = 10 first, getting 5 + 10 ÷ 10 − 3 = 5 + 1 − 3 = 3. Find the error and give the correct answer.10 ÷ 2 = 5, then 5 × 5 = 25
5 + 25 − 3 = 27Each problem shows a student's work. Identify the mistake and find the correct answer.
What is wrong? What is the correct answer?
What is wrong? What is the correct answer?
What is wrong? What is the correct answer?
What is wrong? What is the correct answer?
Identify the error and give the correct answer.
Formulas are expressions that calculate specific quantities. Every formula you encounter in mathematics, science, finance, or engineering depends on the order of operations to give the correct result. If you evaluate a formula in the wrong order, you will calculate the wrong quantity, even if your arithmetic is otherwise perfect. This section applies order of operations to formulas from geometry and science that students have encountered or will encounter throughout this textbook and in later MGU courses.
| Formula | Expression | PEMDAS Step That Matters Most |
|---|---|---|
| Area of trapezoid | A = (1/2)(b₁ + b₂) × h | Parentheses: add the bases first |
| Volume of cylinder | V = π r² h | Exponent: square the radius before multiplying |
| Fahrenheit to Celsius | C = (5/9)(F − 32) | Parentheses: subtract 32 before multiplying by 5/9 |
| Perimeter of rectangle | P = 2(l + w) | Parentheses: add before multiplying by 2 |
| Compound amount | A = P(1 + r)ᵗ | Parentheses then exponent before multiplying by P |
A = (1/2)(b₁ + b₂) × h when b₁ = 6, b₂ = 10, h = 4.Step 1 — Parentheses: 6 + 10 = 16
Step 2 — Multiply: (1/2) × 16 = 8
Step 3 — Multiply h: 8 × 4 = 32C = (5/9)(F − 32).Step 1 — Parentheses: 77 − 32 = 45
Step 2 — Multiply: (5/9) × 45 = 25Step 1 — Exponent: r² = 3² = 9
Step 2 — Multiply: 3.14 × 9 = 28.26
Step 3 — Multiply h: 28.26 × 5 = 141.3P = 2(l + w) when l = 9 and w = 4.A = (1/2) × b × h when b = 8 and h = 5.C = (5/9)(F − 32). What is significant about this temperature?V = π r² h when r = 2, h = 7, and π ≈ 3.14.A = P(1 + r)ᵗ when P = 1000, r = 0.05, t = 2.Answer in complete sentences.
−5² equal −25 and not 25?Show all steps. Rewrite the expression after each step.
3 + 5 × 2(3 + 5) × 24² − 3 × 2 + 110 ÷ 2 × 3 + 42 × 3³ − (4 + 1) × 5100 ÷ (2² × 5) + 3{[2 + 3] × 4 − 6} + 2³48 ÷ [4 × (3 − 1)] + 5(4² − 6) ÷ (5 − 1) + 2 × 33 + [(2 + 1)³ − 5 × 4] ÷ 7(−2)³ + 4−2³ + 4(−3)² − (−3)³6 + (−4) × 2 − 1(−1)¹⁰ × 8 − 3−[5 + (−3)]² + 10(−2)⁴ ÷ 4 × (−3) + 710 − (−2)² + (−2)³(8 + 4) ÷ (3 + 1) + 23 + (2/5) × 10 − 1(3² + 1) ÷ (2 × 5) × 101/2 + (3/4) ÷ (1/4)2.5 + 1.5 × 4(3.6 − 1.6)² ÷ 2 + 0.5(4.8 + 1.2) ÷ 2 × 3 − 4.50.4 × 5 + 6.0 ÷ 1.5 − 0.2Evaluate from the innermost group outward. Show all steps.
3 × [2 + (5 − 3)]{4 × [3 + (2² − 1)] − 2}10 − [2 × (4 + 1) − 3][(6 ÷ 2)² + 1] × 4 − 10{20 ÷ [2 × (3 + 2)] + 3}²2 + {3 × [(5 − 2)² − 4]}Each problem shows a student's work. Identify the specific error, explain why it is wrong, and find the correct answer.
What is wrong? What is the correct answer?
What rule was violated? What is correct?
Identify the error. What is correct?
What grouping was ignored? What is correct?
Name this error. What is the correct answer?
List every error in this work. What is the correct answer?
A = 500 × (1 + 0.04)². Evaluate this expression. Show all steps.C = (5/9)(F − 32) converts Fahrenheit to Celsius. Your oven is set to 350°F. What is that in Celsius?2 + 3 × 4² equals 5 × 16 = 80. Evaluate the expression correctly and explain the student's error.$20 × 3 + $15. Worker B gets $20 × (3 + 1). Who earns more? By how much?[(8 + 4) × (6 + 2)] ÷ 2 tiles. Evaluate this expression and explain what the grouping symbols are doing in context.15 ÷ 5 × 3, multiplication should be performed before division. Explain.−4² not equal (−4)²?3 + 8 × 2 − 4(3 + 8) × 2 − 42⁴ − (5 + 1)² + 10 ÷ 224 ÷ 4 × 3 − 2²(4² − 2 × 3) ÷ (5 − 1) + 1(−3)² + (−3)³6 − (−2)² × 3 + 4(1/3) × 9 + (1/2)² × 40.6 × 5 − 1.5 ÷ 0.5 + 2.1[(−2)³ + 16] ÷ (4 − 2)2 × [10 − (3² + 1)] + 5{4 + [2 × (3 + 1)² − 6]} ÷ 5P = 2(l + w) to find P when l = 12 and w = 7.C = (5/9)(F − 32) to convert 68°F to Celsius.A = (1/2)(b₁ + b₂)h when b₁ = 5, b₂ = 9, and h = 6.3 × [(4 + 2)² − 10] as follows: "First multiply inside: 3 × 4 = 12, then 12 + 2 = 14, then 14² = 196, then 196 − 10 = 186." Identify every error in this work and give the correct answer.Answers may be placed in the answer key.
Order of operations does not disappear when you move into Algebra I. It becomes more important, because algebraic expressions use letters alongside numbers, and every step of simplifying, solving, or evaluating an expression depends on correct order. In Algebra I, you will encounter expressions such as 2x² − 3(x + 4). Evaluating this for a specific value of x requires substitution, applying the exponent, distributing the multiplication across the parentheses, and combining like terms, all in the correct order. You will also work with formulas in science and with function notation, where parentheses and operations appear together constantly. Every skill in this chapter carries forward. If any section felt unclear, return to it before moving on.