3.3 Order of Operations (PEMDAS)

Why Order of Operations Matters

Math is unambiguous only because we agree on the order in which operations run. Without a rule, "3 + 4 × 2" could be 14 (add first) or 11 (multiply first). The agreed order is PEMDAS.

PEMDAS

• Parentheses (and other grouping symbols: brackets [ ], braces { }, absolute-value bars | |, fraction bars)
• Exponents (and roots)
• Multiplication and Division, left to right
• Addition and Subtraction, left to right

The two "left to right" lines are where most people lose points.

Step-by-Step Examples

Example 1: 3 + 4 × 2. Multiplication before addition: 4 × 2 = 8, then 3 + 8 = 11.

Example 2: (3 + 4) × 2. Parentheses first: 3 + 4 = 7, then 7 × 2 = 14. The parentheses change everything.

Example 3: 12 ÷ 3 × 2. Multiplication and division have the same precedence, so go left to right. 12 ÷ 3 = 4, then 4 × 2 = 8. (Doing 3 × 2 first and getting 12 ÷ 6 = 2 is the classic error.)

Example 4: 2 + 3² × 4. Exponent first: 3² = 9. Then multiply: 9 × 4 = 36. Then add: 2 + 36 = 38. (People often do 2 + 3 = 5, then 5² = 25 — wrong. The exponent is on 3 only.)

Example 5: (2 + 3)² × 4. Parentheses first: 2 + 3 = 5. Then exponent: 5² = 25. Then multiply: 25 × 4 = 100.

Grouping Symbols

Anything that groups counts as parentheses: ( ), [ ], { }, | |, and the line of a fraction (it groups the numerator and the denominator separately). Work from the innermost grouping outward.

Example 6: 2 × [3 + (4 − 1)]. Inner parentheses first: 4 − 1 = 3. Now brackets: 3 + 3 = 6. Multiply: 2 × 6 = 12.

Example 7: (8 + 4) ÷ (5 − 2). Both parentheses first: 8 + 4 = 12 and 5 − 2 = 3. Divide: 12 ÷ 3 = 4.

Signed Numbers Inside Order of Operations

Negatives are signs of the numbers, not operators. A leading "−" or a "−" right after a "(" belongs to the next number.

Example 8: 6 + (−2) × 3. The parentheses are just a sign: treat −2 as a number. Multiply: −2 × 3 = −6. Add: 6 + (−6) = 0.

Example 9: −3(4 − 7). Inside parentheses: 4 − 7 = −3. Multiply: −3 × (−3) = 9. Distributing also works: −3 × 4 = −12 and −3 × (−7) = 21; −12 + 21 = 9. Same answer.

Example 10: 10 − 2(3 + 4). Parentheses: 3 + 4 = 7. Multiply: 2 × 7 = 14. Subtract: 10 − 14 = −4. The "10 − 2(7)" is not "8(7) = 56." Multiplication outranks subtraction; multiply 2 × 7 before subtracting.

A Longer Worked Example

Evaluate 3 + 2(4² − 1) − 12 ÷ 3.

1. Parentheses inner: 4² − 1 = 16 − 1 = 15.
2. Multiply: 2 × 15 = 30.
3. Divide: 12 ÷ 3 = 4.
4. Now we have 3 + 30 − 4. Add and subtract left to right: 3 + 30 = 33; 33 − 4 = 29.

Common Errors

1. Doing all multiplication before all division (or all addition before all subtraction). They share precedence — go left to right.
2. Ignoring a negative sign when multiplying. −2 × 3 = −6, not 6.
3. Applying an exponent to a sum. (2 + 3)² = 25, but 2 + 3² = 2 + 9 = 11.
4. Treating "10 − 2(3 + 4)" as "(10 − 2)(3 + 4)." Multiplication next to parentheses is implied; you can't absorb the 2 into the 10.
5. Forgetting the fraction bar as a grouping symbol. In (3 + 5)/(2 + 2), do the top and bottom sums first: 8/4 = 2. It is not 3 + 5/2 + 2.

Calculator Tip

On the NEX you can use a calculator, but it will not save you from entering things in the wrong order. Use parentheses generously to tell the calculator what you mean: type (3 + 4) × 2, not 3 + 4 × 2, if you want the addition done first. The calculator follows PEMDAS too.

More Worked Examples

Evaluate 4 + 2 × 3² − (5 − 1).

1. Parentheses: 5 − 1 = 4.
2. Exponent: 3² = 9.
3. Multiply: 2 × 9 = 18.
4. Now 4 + 18 − 4. Left to right: 4 + 18 = 22; 22 − 4 = 18.

Evaluate 18 − 2(3 + 2²).

1. Inside parentheses, exponent first: 2² = 4.
2. Add inside parentheses: 3 + 4 = 7.
3. Multiply: 2 × 7 = 14.
4. Subtract: 18 − 14 = 4.

Evaluate (−2)³ + 6 ÷ (−2).

1. Exponent: (−2)³ = (−2)(−2)(−2) = −8.
2. Divide: 6 ÷ (−2) = −3.
3. Add: −8 + (−3) = −11.

Evaluate 5 + 2[10 − 3(2 + 1)].

1. Innermost parentheses: 2 + 1 = 3.
2. Multiply inside brackets: 3 × 3 = 9.
3. Subtract inside brackets: 10 − 9 = 1.
4. Multiply by 2: 2 × 1 = 2.
5. Add: 5 + 2 = 7.

Why Implied Multiplication Triples the Error Rate

Expressions like 3(4), 2(x + 1), and 10 − 2(7) all hide a multiplication sign. The NEX loves to test this. Three rules:

• A number directly before parentheses is multiplication: 3(4) = 12.
• A number directly before a variable is multiplication: 2x means 2 × x.
• A negative directly before parentheses distributes: −2(3 + 4) = (−2)(3) + (−2)(4) = −6 + (−8) = −14.

If you treat the implied multiplication as something to do first or skip, every example above silently breaks. Always read it as a times sign.