Order of Operations (PEMDAS)

The rule

When an expression has multiple operations, math has a strict order:

1. Parentheses (and other grouping symbols like brackets, braces).
2. Exponents (including roots).
3. Multiplication and Division — left to right.
4. Addition and Subtraction — left to right.

The acronym PEMDAS captures it. Some places use BODMAS or PEDMAS — same rules.

Critical detail: same-tier operations

Multiplication and division share a tier; do them left to right. Same with addition and subtraction.

$12 \div 4 \times 3$

• Wrong: $12 \div (4 \times 3) = 12 \div 12 = 1$.
• Right: $(12 \div 4) \times 3 = 3 \times 3 = 9$.

Always work left to right within the same tier.

Worked example

$3 + 4 \times 2^2 - (6 - 1)$

1. Parens: $(6 - 1) = 5$, giving $3 + 4 \times 2^2 - 5$.
2. Exponents: $2^2 = 4$, giving $3 + 4 \times 4 - 5$.
3. Multiplication: $4 \times 4 = 16$, giving $3 + 16 - 5$.
4. Add/subtract left to right: $19 - 5 = 14$.

Common pitfalls

• Negatives and exponents. $-3^2$ is $-9$, not $9$. The exponent only applies to the $3$, not the negative sign. To square negative three, write $(-3)^2 = 9$.
• Implicit multiplication. $2(3 + 1)$ means $2 \times 4 = 8$. Evaluate the parentheses first.
• Fraction bars act like parentheses around the top and the bottom. $\dfrac{6 + 4}{2}$ means $\dfrac{10}{2} = 5$, not $6 + 2$.

Key takeaways

• PEMDAS: parens → exponents → ×/÷ → +/−.
• Multiplication and division are equal priority; work left to right. Same for add/subtract.
• $-3^2 \neq (-3)^2$ — watch your negatives.