Order of Operations Formula

Order of operations are the agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right).

The Formula

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

When to use: Without rules, 2+3×42 + 3 \times 4 could mean 20 or 14. We agree to multiply first: 14.

Quick Example

2+3×4=2+12=142 + 3 \times 4 = 2 + 12 = 14 not (2+3)×4=20(2+3) \times 4 = 20

Notation

PEMDAS (or BODMAS): PParentheses, EExponents, MMultiplication/DDivision, AAddition/SSubtraction

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+3×42 + 3 \times 4 could mean 20 or 14. We agree to multiply first: 14.

Formal View

eval(E) is defined recursively: parenthesized sub-expressions first, then ∧, then {×,÷} left-to-right, then {+,−} left-to-right\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+4×23 + 4 \times 2.

Answer

1111

First step

1
Identify the operations present: multiplication and addition.

Full solution

  1. 2
    Apply PEMDAS: multiplication before addition. Compute 4×2=84 \times 2 = 8.
  2. 3
    Now add: 3+8=113 + 8 = 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×(3+5)2−10÷22 \times (3 + 5)^2 - 10 \div 2.

Example 3

medium
Evaluate 23+4×(5−2)2−6÷32^3 + 4 \times (5 - 2)^2 - 6 \div 3 step by step.

Common Mistakes

  • Doing addition before multiplication because it comes first left to right - multiply and divide outrank add and subtract.
  • Treating multiplication as always before division - they share a rank, so go left to right between them.
  • Skipping the innermost parentheses - always resolve the deepest grouping before anything outside it.

Why This Formula Matters

Without a shared order, 2+3×42 + 3 \times 4 could be 14 or 20, and algebra would have no single correct answer. It is the grammar that makes every later expression and equation unambiguous. Recognizing it by "Does this expression have more than one operation that needs an agreed order?" — rather than by familiar numbers — is what lets a student tell it apart from left-to-right reading and distributive property and commutativity in a mixed problem set.

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+3×42 + 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): PParentheses, EExponents, MMultiplication/DDivision, AAddition/SSubtraction

Why is the Order of Operations formula important in Math?

Without a shared order, 2+3×42 + 3 \times 4 could be 14 or 20, and algebra would have no single correct answer. It is the grammar that makes every later expression and equation unambiguous. Recognizing it by "Does this expression have more than one operation that needs an agreed order?" — rather than by familiar numbers — is what lets a student tell it apart from left-to-right reading and distributive property and commutativity in a mixed problem set.

What do students get wrong about Order of Operations?

The procedure for order of operations is the easy part; the trap is doing addition before multiplication because it comes first left to right. Asking "Does this expression have more than one operation that needs an agreed order?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 →
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