Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. The mnemonic is correct as far as it goes. The problem is the cases it does not cover — and those are exactly the cases that generate the viral "what does 8 ÷ 2(2+2) equal?" arguments online.
What PEMDAS gets right
Exponents before multiplication and division, multiplication and division before addition and subtraction. Parentheses override everything. This handles the majority of order-of-operations problems correctly:
3 + 4 × 2 = 3 + 8 = 11 (not 14, which would be wrong). (3 + 4) × 2 = 7 × 2 = 14 (parentheses override).
Where PEMDAS is silent: equal-precedence operations
Multiplication and division have the same precedence. Addition and subtraction have the same precedence. PEMDAS says nothing about what to do when two operations of equal precedence appear in sequence.
The rule is: equal-precedence operations are evaluated left to right.
12 ÷ 3 × 4 = (12 ÷ 3) × 4 = 4 × 4 = 16. Not 12 ÷ 12 = 1.
Students who treat PEMDAS as M before D make this error consistently.
The viral problem: 8 ÷ 2(2+2)
Evaluating the parentheses first: 2+2 = 4. So the expression is 8 ÷ 2(4).
Now the ambiguity: is 2(4) a single term (implicit multiplication with higher precedence than ÷) or is it just 2 × 4 with the same precedence as ÷?
Left-to-right with equal precedence: 8 ÷ 2 × 4 = 4 × 4 = 16. Treating implicit multiplication as higher precedence: 8 ÷ (2×4) = 8 ÷ 8 = 1.
Both readings have defenders. The honest answer is that the expression is ambiguous as written. Professional mathematicians and publications avoid this ambiguity by writing fractions as fractions (8/(2(2+2)) = 1 or (8/2)(2+2) = 16), not as inline division.
The fraction problem
When you write a fraction (numerator over denominator), the fraction bar acts as a grouping symbol. ⁶⁄₂ means 6 ÷ 2 = 3. But if you write 6/2+4 without parentheses, the division symbol does not group the denominator — left-to-right gives (6/2)+4 = 7.
Whenever you intend a fraction with a complex denominator, use parentheses: 6/(2+4) = 1.
Calculators and order of operations
Scientific calculators follow left-to-right with standard precedence, using exactly this convention. TI-84 treats implicit multiplication (juxtaposition) with the same precedence as explicit multiplication. Wolfram Alpha treats it as higher precedence. The same expression can give different answers on different tools — another reason to use explicit parentheses in any expression that is not completely unambiguous.
Frequently Asked Questions
Is PEMDAS the same as BODMAS?
Yes — same rules, different mnemonics. BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) is the UK convention. BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction) is used in Canada. All encode the same mathematical precedence rules.
Does the order of operations apply to algebra?
Yes. In algebraic expressions, the same precedence rules apply: exponents before multiplication, multiplication before addition. The expression 3x + 2 means (3 × x) + 2, not 3 × (x+2). This is why the distributive property requires explicit parentheses to override the default grouping.