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➕ Algebra 1 · Equations

Order of Operations: notes and common mistakes

Study notes

Key ideas

  • The order of operations is an agreement: grouping, then exponents, then multiplication and division, then addition and subtraction.
  • Multiplication and division share a level. Addition and subtraction share a level. Within a level, work from left to right.
  • Grouping symbols include parentheses, brackets, braces, absolute value bars and fraction bars. Do the innermost group first, using the same order inside it.
  • An exponent applies only to what it touches: −a2 means −(a2), but (−a)2 means a2.

Steps

  1. Find the innermost grouping symbol and work it out, using the same order inside.
  2. Evaluate exponents.
  3. Do multiplication and division from left to right.
  4. Do addition and subtraction from left to right.
  5. Rewrite the whole expression after every step, and check that the answer is reasonable.

Formulas

  • The ladder: Grouping, then exponents, then × and ÷ (left to right), then + and − (left to right)
  • Minus sign and exponent: −a2 = −(a2), but (−a)2 = a2
  • Fraction bar: A fraction bar with top a + b and bottom c + d means (a + b) ÷ (c + d), when c + d ≠ 0

Watch out

  • PEMDAS is not a six-step staircase. M and D tie, and A and S tie, so work each pair left to right: 15 − 9 + 2 = 8, not 15 − 11 = 4.
  • Going left to right through every symbol is not the rule. Only operations on the same level go left to right.
  • The exponent touches only the number right before it, so 2 × 32 = 18, not 36.
  • A fraction bar groups its top and its bottom. When you type it on one line, add parentheses.
  • When you substitute a negative number for a variable, wrap it in parentheses.
  • If a group in a divisor works out to 0, the expression is undefined.

Why it works

  • Why have a rule at all? Without one, 3 + 4 × 5 would have two answers. A shared rule lets people (and most calculators and software) agree.
  • Why does the ranking look this way? Each level is a shortcut for repeating the level below it: for whole numbers, multiplication is repeated addition and an exponent is repeated multiplication. With these ranks you can write 2x2 + 3x + 1 with no parentheses at all.

Common mistakes

✗ MistakePEMDAS means multiplication always happens before division, and addition always before subtraction.
✓ InsteadM and D tie, and A and S tie. Work each tie left to right: 27 ÷ 3 × 3 = 9 × 3 = 27, not 27 ÷ 9 = 3.
✗ MistakeJust work left to right: 7 + 2 × 5 = 45.
✓ InsteadOnly ties go left to right. Multiplication ranks above addition, so 2 × 5 = 10 first, and then 7 + 10 = 17.
✗ Mistake−112 = 121.
✓ InsteadThe exponent touches only the 11, so −112 = −121. Write (−11)2 when you want 121.
✗ MistakeYou can type the fraction with top 8 + 4 and bottom 2 + 1 as 8 + 4 ÷ 2 + 1.
✓ InsteadThat gives 8 + 2 + 1 = 11. The bar groups the top and the bottom, so type (8 + 4) ÷ (2 + 1) = 12 ÷ 3 = 4.
✗ MistakeWhen x = −10, x2 is −102 = −100.
✓ InsteadSubstitute with parentheses: x2 = (−10)2 = 100.

Teach it back

Explain to a friend (or a stuffed animal) how to evaluate 8 + 18 ÷ (5 − 2) × 3 and why each step happens in that order.

Explaining it out loud shows what you really know. Did you…

  • I did the parentheses first: 5 − 2 = 3.
  • I said that division and multiplication tie, so I went left to right: 18 ÷ 3 = 6, then 6 × 3 = 18.
  • I added last and got 8 + 18 = 26.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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