Order of Operations

Order of Operations

When an expression has several operations mixed together, everyone needs to agree on the order they're performed in — otherwise 2 + 3 \times 4 could mean either 20 or 14. The agreed-upon order is often remembered with the acronym PEMDAS (or GEMDAS): Parentheses / Grouping, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

The Rules in Detail

1. Grouping symbols first. Parentheses (), brackets [\,], and fraction bars all tell you to simplify what's inside them before anything else. If groups are nested, work from the innermost outward.
2. Exponents next. Evaluate powers and roots before multiplying, dividing, adding, or subtracting.
3. Multiplication and division together, left to right. These two operations have equal priority — do not always multiply before dividing. Whichever comes first when reading left to right happens first.
4. Addition and subtraction together, left to right. Just like multiplication and division, these share equal priority and are done in reading order.

Nested Parentheses and Exponents

For nested grouping symbols like 2\big[5 + (3^2 - 4)\big], always resolve the innermost parentheses first: 3^2 - 4 = 9 - 4 = 5, so the expression becomes 2[5+5] = 2 \times 10 = 20. When an exponent applies to something in parentheses, such as (-2)^4, the base includes everything inside the parentheses — here that means (-2)^4 = 16 (a positive result, since 4 is even).

Common Mistakes

• Sign errors with exponents: -2^4 and (-2)^4 are not the same! Without parentheses, the exponent binds only to the 2, so -2^4 = -(2^4) = -16, while (-2)^4 = 16.
• Treating multiplication as always before division (or addition always before subtraction): remember these pairs are evaluated left to right together, not in two separate passes.
• Implicit multiplication: an expression like 3(4+5) or 2x means multiplication even though there's no \times symbol; don't forget to distribute or multiply it in.
• Forgetting the fraction bar acts as grouping: in \frac{3+5}{2^2}, both the numerator 3+5 and the denominator 2^2 must be simplified separately before dividing.

Example

Evaluate 5 + 2 \times (8 - 3)^2 \div 5.

Step 1 (innermost parentheses): 8 - 3 = 5, giving 5 + 2 \times 5^2 \div 5.

Step 2 (exponent): 5^2 = 25, giving 5 + 2 \times 25 \div 5.

Step 3 (multiplication/division, left to right): 2 \times 25 = 50, then 50 \div 5 = 10, giving 5 + 10.

Step 4 (addition): 5 + 10 = 15.

So 5 + 2 \times (8-3)^2 \div 5 = 15.

Key Takeaways

• Follow PEMDAS/GEMDAS: grouping symbols, then exponents, then multiplication/ division left to right, then addition/subtraction left to right.
• Multiplication and division are equal priority (same for addition and subtraction) — always work left to right within each pair.
• Watch out for -a^n vs. (-a)^n, implicit multiplication, and fraction bars that act as grouping symbols.