Order of Operations Formula

The Formula

Parentheses \to Exponents \to Multiplication/Division (left to right) \to Addition/Subtraction (left to right)

When to use: Without rules, 2 + 3 \times 4 could mean 20 or 14. We agree to multiply first: 14.

Quick Example

2 + 3 \times 4 = 2 + 12 = 14 not (2+3) \times 4 = 20

Notation

PEMDAS (or BODMAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

What This Formula Means

The agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Without rules, 2 + 3 \times 4 could mean 20 or 14. We agree to multiply first: 14.

Formal View

\text{eval}(E) \text{ is defined recursively: parenthesized sub-expressions first, then } \wedge, \text{ then } \{\times, \div\} \text{ left-to-right, then } \{+, -\} \text{ left-to-right}

Worked Examples

Example 1

easy
Evaluate 3 + 4 \times 2.

Solution

  1. 1
    Identify the operations present: multiplication and addition.
  2. 2
    Apply PEMDAS: multiplication before addition. Compute 4 \times 2 = 8.
  3. 3
    Now add: 3 + 8 = 11.

Answer

11
The order of operations (PEMDAS/BODMAS) tells us to perform multiplication before addition, even when addition appears first in the expression.

Example 2

medium
Evaluate 2 \times (3 + 5)^2 - 10 \div 2.

Common Mistakes

  • Doing operations left to right without considering precedence
  • Forgetting parentheses first

Why This Formula Matters

Required for unambiguous communication in math and programming.

Frequently Asked Questions

What is the Order of Operations formula?

The agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

How do you use the Order of Operations formula?

Without rules, 2 + 3 \times 4 could mean 20 or 14. We agree to multiply first: 14.

What do the symbols mean in the Order of Operations formula?

PEMDAS (or BODMAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

Why is the Order of Operations formula important in Math?

Required for unambiguous communication in math and programming.

What do students get wrong about Order of Operations?

Forgetting that \times and \div have equal precedence (go left to right).

What should I learn before the Order of Operations formula?

Before studying the Order of Operations formula, you should understand: addition, subtraction, multiplication, division.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Commutative, Associative, and Distributive Properties →
← Back to Order of Operations See All Examples Practice Problems