Common mistakes with order of operations and left-to-right evaluation
Reserve & Extensions • K-12
You probably know PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. But knowing the acronym and applying it correctly are two very different things. This lesson focuses on the tricky cases where even careful students make mistakes.
| Priority | Operations | Rule |
|---|---|---|
| 1st | Parentheses (and other grouping) | Innermost first |
| 2nd | Exponents | Evaluate powers and roots |
| 3rd | Multiplication and Division | Left to right (equal priority) |
| 4th | Addition and Subtraction | Left to right (equal priority) |
PEMDAS does not mean multiplication comes before division. They have equal priority and are performed left to right.
Evaluate: 24 ÷ 4 × 3
Wrong approach: Doing multiplication first would give 24 ÷ 12 = 2. That is incorrect.
There is a critical difference between -32 and (-3)2:
The exponent applies only to 3. The negative is applied after.
The parentheses make the negative part of the base.
Evaluate: -52 + 10
Common wrong answer: 35 (from incorrectly computing (-5)2 = 25, then 25 + 10).
A fraction bar acts like parentheses around the entire numerator and the entire denominator.
Evaluate: (8 + 4) / (2 + 1)
If you ignored grouping and wrote 8 + 4 / 2 + 1, you would get 8 + 2 + 1 = 11. Very different!
When you see 2(3 + 4), the multiplication between 2 and the parentheses is not given higher priority than other multiplication or division. It follows normal left-to-right rules.
Expressions like "8 ÷ 2(2+2)" go viral because they are ambiguously written. In standard mathematical convention, multiplication and division are left to right: 8 ÷ 2 × 4 = 4 × 4 = 16. The real lesson: use parentheses to make your meaning clear. Good notation prevents confusion.
Evaluate: 10 - 3 + 2
Incorrect: doing addition first gives 10 - 5 = 5.
If you are ever unsure about order of operations, add parentheses to force the order you intend. Clear notation is not a sign of weakness -- it is a sign of precision.
1. Evaluate: 18 ÷ 6 × 3
Left to right: 18 ÷ 6 = 3, then 3 × 3 = 9.
2. Evaluate: -42
The exponent applies to 4 only: -(42) = -16. (Not 16!)
3. Evaluate: 3 + 4 × 2 - 1
Multiplication first: 4 × 2 = 8. Then left to right: 3 + 8 - 1 = 10.
4. Evaluate: (6 + 2)2 ÷ 4 - 3
Parentheses: 8. Exponent: 82 = 64. Division: 64 ÷ 4 = 16. Subtraction: 16 - 3 = 13.
5. Evaluate: 2 × 32 - 4(5 - 2)
Parentheses: 5 - 2 = 3. Exponent: 32 = 9. Multiplication: 2 × 9 = 18 and 4 × 3 = 12. Subtraction: 18 - 12 = 6.