Order of Operations

7 min read

Textbook Reference

This source is free and openly licensed. The order-of-operations rule is presented in Section 1.1 alongside the real numbers.

Key idea

The order of operations is not an arbitrary rule someone invented to trip you up. It is a shared agreement that lets a single written expression mean exactly one number.

Consider $2 + 3 \times 4$. If you add first you get $20$; if you multiply first you get $14$. Both are reasonable readings of the symbols, and that is the problem. Mathematics cannot allow one expression to have two answers, so the world agreed on a fixed reading order: do multiplication before addition. With the agreement in place, $2 + 3 \times 4$ has one meaning, $14$, everywhere on Earth.

This means: the order of operations is a parsing rule, the grammar of arithmetic. It is the same idea as a programming language deciding that 2 + 3 * 4 evaluates the multiplication first. Learn the grammar once and every expression reads unambiguously.

Prerequisite Check

Before this lesson, make sure you can do all of the following:

• Add, subtract, multiply, and divide real numbers, including negatives
• Recognize an exponent: $2^3 = 2 \cdot 2 \cdot 2 = 8$
• Tell the difference between grouping symbols: parentheses $(\ )$, brackets $[\ ]$, and braces $\{\ \}$
• Understand that a fraction bar groups everything above it and everything below it

If exponents are unfamiliar, the Integer Exponent Rules lesson covers them; a quick reminder appears below.

Quick Reference

The order of operations, highest priority first.

The common acronym. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

The single most-missed point. Multiplication and division share one priority level; addition and subtraction share another. Within a shared level you work strictly left to right. The “MD” in PEMDAS does not mean multiplication always comes before division.

Key Concepts

1. Why the Order Exists

An expression is a string of numbers and operation symbols. Without a fixed reading order, the same expression could produce different numbers. The order of operations removes that ambiguity so every expression names one value.

Example 1. Evaluate $2 + 3 \times 4$ correctly.

Multiplication has higher priority than addition, so multiply first: \[ 2 + 3 \times 4 = 2 + 12 = 14 \]

The reading “add first to get $20$” violates the agreement and is wrong.

2. Grouping Symbols Come First, From the Inside Out

Anything inside parentheses, brackets, or braces is evaluated before it interacts with the rest of the expression. When groups are nested, work the innermost group first.

Example 2. Evaluate $3 \times (4 + 2)$.

\[ 3 \times (4 + 2) = 3 \times 6 = 18 \]

The parentheses change the answer. Without them, $3 \times 4 + 2 = 14$.

Example 3. Evaluate $2 \times [3 + (5 - 1)]$.

Work the innermost group first, then the bracket: \[ 2 \times [3 + (5 - 1)] = 2 \times [3 + 4] = 2 \times 7 = 14 \]

Important: a fraction bar is a grouping symbol. In $\dfrac{8 + 4}{2 + 1}$, the bar groups the whole top and the whole bottom. Evaluate each separately, then divide: $\dfrac{12}{3} = 4$. Writing $8 + \frac{4}{2} + 1$ instead is a different expression with a different value.

3. Exponents Before Multiplication

After grouping, evaluate exponents and roots, then move to multiplication and division.

Example 4. Evaluate $5 \times 2^3$.

The exponent acts before the multiplication: \[ 5 \times 2^3 = 5 \times 8 = 40 \]

Important: an exponent attaches only to what it touches. In $-3^2$, the exponent touches the $3$, not the negative sign, so $-3^2 = -(3^2) = -9$. To square the negative number, write $(-3)^2 = 9$ with parentheses. This single distinction causes a large share of sign errors.

4. Multiplication and Division: Equal Priority, Left to Right

Multiplication and division are done in the order they appear, scanning left to right. Neither one automatically wins.

Example 5. Evaluate $24 \div 4 \times 2$.

Scan left to right. The division appears first, so do it first: \[ 24 \div 4 \times 2 = 6 \times 2 = 12 \]

A student who multiplies first gets $24 \div 8 = 3$, which is wrong. Left to right is the deciding rule.

5. Addition and Subtraction: Equal Priority, Left to Right

The same left-to-right rule governs addition and subtraction, which are the lowest priority.

Example 6. Evaluate $10 - 4 + 3$.

\[ 10 - 4 + 3 = 6 + 3 = 9 \]

Doing the addition first would give $10 - 7 = 3$, which is wrong. Subtraction and addition are taken in the order written.

6. A Full Worked Example

Example 7. Evaluate $4 + 3 \times (6 - 2)^2 \div 8$.

Step 1 (grouping): $6 - 2 = 4$, giving $4 + 3 \times 4^2 \div 8$.

Step 2 (exponent): $4^2 = 16$, giving $4 + 3 \times 16 \div 8$.

Step 3 (multiply and divide, left to right): $3 \times 16 = 48$, then $48 \div 8 = 6$, giving $4 + 6$.

Step 4 (add): $4 + 6 = 10$.

\[ 4 + 3 \times (6 - 2)^2 \div 8 = 10 \]

Common Errors Summary

Common Misconceptions

Leveled Practice

Level 1: Direct Application

Problem 1. Evaluate $7 + 2 \times 5$.

Problem 2. Evaluate $(8 - 3) \times 2$.

Problem 3. Evaluate $36 \div 6 \div 2$.

Level 2: Multiple Steps

Problem 4. Evaluate $2 + 3 \times 4^2 - 10$.

Problem 5. Evaluate $\dfrac{12 + 6}{3} - 2^2$.

Level 3: Deeper Reasoning

Problem 6. Evaluate $5 - 2[3 + (8 - 6)^2]$.

Problem 7. A classmate writes $-4^2 = 16$. Explain the error and give the correct value.

Mastery Checklist

You have mastered this skill when you can do all of the following without referring to notes:

• Evaluate an expression using the full priority order: grouping, exponents, multiply or divide, add or subtract
• Work nested grouping symbols from the inside out
• Apply the left-to-right rule within multiplication and division
• Apply the left-to-right rule within addition and subtraction
• Treat a fraction bar as a grouping symbol
• Distinguish $-a^2$ from $(-a)^2$

Mental Model

Think of an expression as a building you tear down in layers, deepest first.

The deepest layer is whatever is wrapped in grouping symbols; collapse those to single numbers before anything else. Then handle exponents, which bind tighter than ordinary multiplication. Then sweep left to right doing multiplications and divisions. Finally sweep left to right doing additions and subtractions. At each layer you replace a chunk of the expression with one number, and the expression gets shorter until a single value remains.

The acronym PEMDAS names the layers, but remember the two “ties”: multiply and divide share a layer, and add and subtract share a layer. In a tie, the leftmost operation goes first.

Connections

Within MATH 114

• Integer exponents: the exponent layer of the order of operations is exactly where the exponent rules are applied.
• Simplifying expressions: combining like terms and distributing both rely on a correct order of operations.
• Solving linear equations: to undo an expression you reverse the order of operations, the so-called “unwrapping” of an equation.

Toward Functions

• Evaluating functions: substituting a number into a function and simplifying is one long order-of-operations exercise.

Audience Notes

For students who find math intimidating: This is a checklist, not a mystery. Tear the expression down one layer at a time and never skip ahead. Slow and ordered beats fast and wrong.

For students interested in proof: The order of operations is a human convention, not a theorem. Different conventions are logically possible; the field simply standardized on this one so written mathematics is unambiguous.

For students interested in careers: Every programming language and spreadsheet enforces an operator-precedence rule that mirrors PEMDAS. A misread precedence is a real and common software bug, which is why programmers add parentheses for clarity even when they are not strictly required.

Back to MATH 114 Skills | Previous: The Real Number System | Next: Integer Exponent Rules