Order of Operations with Exponents

SECTION 1

Where Did the Order of Operations Come From?

Imagine texting a friend the problem 3 + 2 × 4. Your friend gets 20, but you get 11. Who's right? For centuries, mathematicians ran into this exact confusion. They needed a set of rules so that everyone would get the same answer every time. That's why the order of operations was created—a universal set of instructions for simplifying math expressions.

1500s

Early Notation Takes Shape

European mathematicians begin using symbols like + and − instead of writing words. This makes formulas shorter but creates new questions about which operation to do first.

1600s

Exponent Notation Appears

René Descartes introduces the small raised number we now call an exponent. Writing x² is much easier than writing x × x every time.

1800s

Rules Become Standard

Textbooks around the world begin teaching that multiplication and division come before addition and subtraction, and exponents come before multiplication.

1900s

PEMDAS Enters Classrooms

American schools popularize the memory aid PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) so students can remember the correct order.

Without these rules, the same expression could give different answers depending on who reads it. The order of operations is like a universal math language that keeps everyone on the same page. So how exactly do exponents fit into this system? Let's find out.

SECTION 2

Core Principles & Key Definitions

Before we dive into examples, let's nail down the vocabulary. An exponent (sometimes called a "power") is the small number written above and to the right of a base. It tells you how many times to multiply the base by itself. For example, 2³ means 2 × 2 × 2, which equals 8.

Parentheses First

Always start inside parentheses (or brackets). Simplify everything inside them before moving on.

Exponents Second

After parentheses, evaluate all exponents. This means computing powers like 3² = 9 or 5³ = 125.

Multiply & Divide Third

Work multiplication and division from left to right. They share the same priority level—neither one always comes first.

Add & Subtract Last

Finally, handle addition and subtraction from left to right. Like multiplication and division, they share a priority level.

💡 PEMDAS Reminder

Use the phrase "Please Excuse My Dear Aunt Sally" to remember: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction. The M and D share a step (left to right), and so do A and S.

✦ KEY TAKEAWAY

Think of the order of operations like a recipe. You have to crack the eggs (parentheses) before you whisk them (exponents), and you whisk before you pour the batter into the pan (multiply/divide), and you bake before you frost (add/subtract). Skipping a step or doing them out of order ruins the cake—and gives you the wrong answer.

SECTION 3

Visualizing the Order of Operations

The diagram below shows the order of operations as a staircase. You always start at the top and work your way down. Each step must be completed before you move to the next one.

The staircase shows the four levels of priority. Notice that Exponents sit on Step 2, right after Parentheses. You must evaluate every exponent before you multiply, divide, add, or subtract.

Look at the arrow on the right side of the diagram. It runs from "Highest Priority" to "Lowest Priority." Parentheses have the highest priority, so you handle them first. Exponents come right after. Many students forget this step and jump straight to multiplication. Don't skip the staircase!

SECTION 4

The Math Behind Exponents

An exponent is a shortcut for repeated multiplication. Instead of writing 5 × 5 × 5, you can write 5³. The base is 5 and the exponent is 3. Let's formalize this.

EXPONENT DEFINITION

bⁿ = b × b × b × … × b (n times)

Here b is the base (the number being multiplied) and n is the exponent (how many times you multiply the base by itself). The exponent must be a whole number (0, 1, 2, 3, …).

SPECIAL CASE — EXPONENT OF 1

b¹ = b

Any number raised to the first power equals itself. For example, 7¹ = 7.

SPECIAL CASE — EXPONENT OF 0

b⁰ = 1 (when b ≠ 0)

Any nonzero number raised to the zero power equals 1. So 10⁰ = 1, 999⁰ = 1, and even 2⁰ = 1.

⚠️ WATCH OUT

Be careful with negative signs! The expression −3² means −(3²) = −9 because the exponent only applies to the 3. If you want the negative sign included, you need parentheses: (−3)² = (−3) × (−3) = 9.

Now let's see how exponents fit into a full expression. When you see something like 4 + 3² × 2, you follow PEMDAS: there are no parentheses, so handle the exponent first (3² = 9), then multiply (9 × 2 = 18), and finally add (4 + 18 = 22).

SECTION 5

Step-by-Step Breakdown of Common Patterns

Let's look at the most common types of expressions you'll encounter. The table below shows each pattern and how to handle it correctly.

Common expression patterns with exponents

Expression	Step-by-Step Work	Final Answer
2³ + 5	= 8 + 5	13
10 − 2⁴	= 10 − 16	−6
3 × 2² + 1	= 3 × 4 + 1 = 12 + 1	13
(1 + 3)² × 2	= 4² × 2 = 16 × 2	32
5² − (6 + 2) ÷ 4	= 25 − 8 ÷ 4 = 25 − 2	23

This flowchart walks through 3 × (2 + 1)² − 5 one step at a time. Each colored box represents one priority level of PEMDAS. Notice how the expression gets simpler at every step until only the final answer remains.

Notice the pattern: every step replaces one operation with its result. The expression gets shorter each time. By the end, you're left with a single number. That's how you know you're done!

SECTION 6

Worked Example

Let's work through a slightly trickier expression together. We'll evaluate 2⁴ + 3 × (10 − 6)² ÷ 8. Follow along step by step.

Evaluate: 2⁴ + 3 × (10 − 6)² ÷ 8

Step 1 — ParenthesesLook for parentheses first. We see (10 − 6). Simplify: 10 − 6 = 4. The expression becomes 2⁴ + 3 × 4² ÷ 8.

2⁴ + 3 × 4² ÷ 8

Step 2 — ExponentsNow handle all exponents from left to right. 2⁴ = 2 × 2 × 2 × 2 = 16. Then 4² = 4 × 4 = 16. Replace both.

16 + 3 × 16 ÷ 8

Step 3 — Multiplication and Division (left to right)Moving left to right, we see 3 × 16 first. That equals 48. Now the expression is 16 + 48 ÷ 8. Next we have 48 ÷ 8 = 6.

16 + 6

Step 4 — AdditionFinally, add 16 + 6 = 22.

⚠️ COMMON MISTAKE

Some students add 16 + 3 before multiplying. That gives 19 × 16 ÷ 8, which leads to a wrong answer. Remember: multiplication and division always come before addition and subtraction.

SECTION 7

Common Mistakes & How to Avoid Them

Even great math students make mistakes with order of operations. Knowing the most common errors ahead of time helps you avoid them. The table below shows what goes wrong and how to fix it.

Common order-of-operations errors

Common Mistake	Why It's Wrong	Correct Approach
Working left to right, ignoring priority	3 + 2 × 4 ≠ 20. You can't just go left to right for everything.	Multiply first: 2 × 4 = 8, then add: 3 + 8 = 11.
Skipping exponents	Treating 2³ as 2 × 3 = 6 instead of 2 × 2 × 2 = 8.	An exponent means repeated multiplication of the base, not base × exponent.
Thinking M always comes before D	In PEMDAS, M and D share the same step. 12 ÷ 3 × 2 = 8, not 2.	Do multiplication and division in the order they appear, left to right.
Forgetting parentheses change the base	−5² = −25 but (−5)² = 25. The parentheses matter!	Check whether the negative sign is inside or outside the parentheses.

✦ KEY TAKEAWAY

Think of PEMDAS like a video-game level system. Parentheses are Level 1 — you must clear them first. Exponents are Level 2 — you can't skip ahead to Level 3 (multiply/divide) without finishing Level 2. If you rush through levels out of order, you'll get a "Game Over" (wrong answer). Take it one level at a time!

SECTION 8

Connecting to Algebra and Beyond

Everything you've learned about order of operations with numbers works exactly the same way when you start using variables. In algebra, you'll see expressions like 3x² + 2x − 5 and you'll need to evaluate them by plugging in a number for x and then following PEMDAS.

From numerical expressions to algebraic expressions

What You Know Now	What's Coming Next
Evaluate 3² + 4 = 13	Evaluate x² + 4 when x = 3 → 3² + 4 = 13
Whole-number exponents (2, 3, 4, …)	Negative and fractional exponents (2⁻¹ = ½)
Numerical expressions only	Writing and solving equations with exponents
Single exponents like 5³	Exponent rules: x² × x³ = x⁵

The skills you're building right now are the foundation for every math class ahead. Once PEMDAS is second nature, you'll be able to tackle algebra, geometry formulas, and even scientific equations with confidence.

SECTION 9

Practice Problems

Try these five problems on your own. Work through each one step by step before checking the answer. The problems get harder as you go!

PROBLEM 1 — CONCEPTUAL

In the expression 5 + 2³ × 4, which operation should you perform first: the addition, the exponent, or the multiplication? Explain why.

PROBLEM 2 — BASIC CALCULATION

Evaluate: 4² − 3 × 2 + 1

PROBLEM 3 — INTERMEDIATE

Evaluate: (5 − 2)³ + 4 × 2² ÷ 8

PROBLEM 4 — APPLIED

A square garden has sides of length 5 feet. A triangular section with base 4 feet and height 3 feet is cut out. Write a numerical expression for the remaining area and evaluate it. (Hint: Area of a square = s². Area of a triangle = b × h ÷ 2.)

PROBLEM 5 — CRITICAL THINKING

Using only the numbers 2, 3, and 4 (each used exactly once), parentheses, addition, and an exponent, write an expression that equals 11. Then write a different expression using the same numbers and operations that equals 81.

SUMMARY

Lesson Summary

The order of operations is a set of rules that tells you which part of an expression to simplify first. The memory aid PEMDAS stands for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right). An exponent tells you how many times to multiply the base by itself, and it always gets evaluated right after parentheses—before any multiplication, division, addition, or subtraction.

To evaluate an expression correctly, work through each priority level one at a time. Simplify inside parentheses first, then evaluate exponents, then perform multiplication and division from left to right, and finally handle addition and subtraction from left to right. Watch out for common traps like confusing 2³ with 2 × 3, or forgetting that −5² and (−5)² give different results. These skills are the foundation for all of algebra!