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TACHS Math • Number Sense and Operations

Use order of operations to evaluate expressions

Learn the exact steps that guarantee every math expression gives one correct answer.

Explore the lesson 9 min read
TACHS illustration with a Catholic high school, reading book, and math calculator
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Why Do We Need Rules for Math Order?

Imagine texting a friend: "Solve 3 + 4 × 2." Your friend says 11. You say 14. Who is right? Without a shared set of rules, the same expression can give different answers. That is why mathematicians created the order of operations — a set of rules everyone agrees to follow so that every expression has exactly one correct value.

These rules did not appear overnight. They developed over centuries as math became more complex. Let's look at how they came to be.

1500s • Early Algebra Takes Shape

European mathematicians began writing equations with symbols like + and −. They needed rules so everyone would read the same expression the same way.

1600s • Exponents Appear

René Descartes introduced the small raised numbers we call exponents. Mathematicians agreed that exponents should be calculated before multiplication.

1800s • Textbooks Standardize the Rules

As schools grew, math textbooks printed a fixed order: parentheses first, then exponents, then multiplication and division, then addition and subtraction.

1900s • PEMDAS Becomes a Classroom Staple

Teachers in the United States popularized the memory trick PEMDAS ("Please Excuse My Dear Aunt Sally") to help students remember the order.

So the big question is: In what exact order do you perform operations to get the one correct answer? That is what this lesson will teach you, step by step.

SECTION 2

Core Principles of the Order of Operations

The order of operations is often remembered by the acronym PEMDAS. Each letter stands for a step you follow, from first to last. Let's break it down.

1

P — Parentheses

Always start inside parentheses (or brackets). Work from the innermost set outward.
2

E — Exponents

Next, evaluate any exponents (powers). For example, 3² means 3 × 3 = 9.
3

MD — Multiply & Divide

Work from left to right. Multiplication and division are equals — do whichever comes first as you read left to right.
4

AS — Add & Subtract

Finally, work from left to right. Addition and subtraction are equals — do whichever comes first as you read left to right.
⚠

Common Mistake Alert

Many students think multiplication always comes before division, or addition always comes before subtraction. That is not true! Multiplication and division are done left to right, and addition and subtraction are done left to right. Think of M and D as a tied pair, and A and S as another tied pair.
✦

KEY TAKEAWAY

Think of PEMDAS like getting dressed in the morning. You have to put on your socks before your shoes, and your shirt before your jacket. If you do things out of order, you get a weird result. Math works the same way — the right order gives you the right answer every time.

SECTION 3

Seeing the Order of Operations

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

PEMDAS Staircase — Work Top to BottomPParenthesesDo everything inside ( ) first, innermost first.EExponentsEvaluate powers like 2³ = 8.MDMultiply & DivideLeft to right — whichever comes first.ASAdd & SubtractLeft to right — whichever comes first.Final Answer
Start at the top with Parentheses and work your way down. Notice that Multiply/Divide and Add/Subtract are each on the same step because they share the same priority.

Notice how the staircase has four levels, not six. That is because Multiply and Divide share one level, and Add and Subtract share another. Within a shared level, you simply move left to right through the expression.

SECTION 4

The PEMDAS Framework in Action

Let's see the rules written out as clear steps. When you look at any expression, follow these steps one at a time.

STEP 1 — PARENTHESES

Simplifyeverythinginside()or[]firstSimplify everything inside ( ) or [ ] first

If parentheses are nested, start with the innermost pair. For example, in 2 × [3 + (4 − 1)], solve (4 − 1) = 3 first.

STEP 2 — EXPONENTS

Evaluateallpowers:anmeansa×a×...×a(ntimes)Evaluate all powers: aⁿ means a × a × ... × a (n times)

For example, 5² = 5 × 5 = 25 and 2³ = 2 × 2 × 2 = 8.

STEP 3 — MULTIPLY & DIVIDE (LEFT TO RIGHT)

a×bora÷ba × b or a ÷ b

Scan the expression from left to right. Perform each × or ÷ in the order you meet it. Example: 12 ÷ 3 × 2 → first 12 ÷ 3 = 4, then 4 × 2 = 8.

STEP 4 — ADD & SUBTRACT (LEFT TO RIGHT)

a+bora−ba + b or a − b

Scan from left to right again. Perform each + or − in order. Example: 10 − 3 + 5 → first 10 − 3 = 7, then 7 + 5 = 12.

💡

Quick Tip

When you see a fraction bar, treat the top (numerator) and bottom (denominator) as if they are each inside their own parentheses. Simplify each part separately, then divide.

SECTION 5

Breaking Down a Multi-Step Expression

Let's trace every move through the expression 4 + 3 × (10 − 2²). The diagram below highlights which part of the expression you handle at each step.

Step-by-Step: 4 + 3 × (10 − 2²)STEP 1 — Exponent inside parentheses4 + 3 × (10 − 2²)→ 2² = 4Expression becomes: 4 + 3 × (10 − 4)STEP 2 — Subtraction inside parentheses4 + 3 × (10 − 4)→ 10 − 4 = 6Expression becomes: 4 + 3 × 6STEP 3 — Multiplication4 + 3 × 6→ 3 × 6 = 18Expression becomes: 4 + 18STEP 4 — Addition4 + 18→ 4 + 18 = 22Final Answer = 22
Each colored box shows the part of the expression being evaluated at that step. Notice how the exponent inside the parentheses is handled before the subtraction inside those same parentheses, because Exponents come before Subtraction in PEMDAS.

Inside the parentheses, you still follow PEMDAS. That is why we handled the exponent 2² before the subtraction 10 − 4. Once the parentheses are cleared, you move on to multiplication, and then addition.

SECTION 6

Worked Example

Let's evaluate the expression 18 ÷ 3² + 5 × (7 − 4) step by step.

Evaluate: 18 ÷ 3² + 5 × (7 − 4)

  1. 1

    Step 1 — Parentheses

    Look for parentheses. We see (7 − 4). Simplify inside: 7 − 4 = 3.
    Expression becomes: 18 ÷ 3² + 5 × 3
  2. 2

    Step 2 — Exponents

    Look for exponents. We see 3². Calculate: 3² = 3 × 3 = 9.
    Expression becomes: 18 ÷ 9 + 5 × 3
  3. 3

    Step 3 — Multiply & Divide (left to right)

    Scan left to right. The first operation at this level is 18 ÷ 9 = 2. Continue scanning: 5 × 3 = 15.
    Expression becomes: 2 + 15
  4. 4

    Step 4 — Add & Subtract (left to right)

    Only addition remains. 2 + 15 = 17.
    Final Answer: 17
💡

Check Your Work

A great habit is to rewrite the expression after each step. This helps you avoid accidentally skipping an operation or mixing up numbers.

SECTION 7

Common Errors and How to Avoid Them

Even when students know PEMDAS, they sometimes make mistakes. Here are the most common errors and the correct approach side by side.

Common ErrorWhy It's WrongCorrect Approach
Adding before multiplying: 3 + 4 × 2 = 14Multiplication comes before addition in PEMDAS.4 × 2 = 8 first, then 3 + 8 = 11.
Always multiplying before dividing: 12 ÷ 4 × 3 = 1M and D have equal priority. Work left to right.12 ÷ 4 = 3 first (left), then 3 × 3 = 9.
Forgetting exponents: 2 + 3² = 5² = 25The exponent applies only to the base right next to it, not to 2 + 3.3² = 9 first, then 2 + 9 = 11.
Ignoring nested parentheses: [2 × (1 + 3)]² = [2 × 4]² but student writes 2 × 4² = 32The brackets group 2 × 4 together. The exponent applies to the whole bracket.[2 × (1 + 3)] = [2 × 4] = 8, then 8² = 64.

Common order-of-operations mistakes and corrections

✦

KEY TAKEAWAY

Think of "left to right" as reading a book. You don't skip to the middle of a sentence — you read word by word from left to right. Do the same with multiplication/division and addition/subtraction: handle each operation in the order you meet it.

SECTION 8

From Numbers to Algebra and Beyond

Right now, you are using the order of operations with plain numbers. In algebra, you will use the same rules with variables (letters that stand for numbers). The rules never change — PEMDAS works the same way whether you see 3 + 4 × 2 or 3 + 4x.

What You Learn NowHow It Connects Later
Evaluate 5 + 2 × 3Simplify 5 + 2x when x = 3
Parentheses change the orderGrouping symbols in algebraic expressions and equations
Exponents with numbers: 4²Exponents with variables: x², polynomial expressions
Left-to-right rule for ×, ÷Solving multi-step equations with inverse operations

How order of operations connects to future math topics

Mastering the order of operations now builds a strong foundation. Every equation you solve in algebra, geometry, and even calculus will rely on these same rules. You are learning the grammar of math!

SECTION 9

Practice Problems

Try each problem on your own before reading the answer. Remember: write the expression, then work through PEMDAS one step at a time.

Question 1: CONCEPTUAL

In the expression 6 + 2 × 5, which operation should you perform first: the addition or the multiplication? Explain why.
Show answer
You should perform the multiplication first. According to PEMDAS, multiplication comes before addition. So you calculate 2 × 5 = 10 first, and then add 6 + 10 = 16. If you added first, you would get 8 × 5 = 40, which is incorrect.

Question 2: BASIC CALCULATION

Evaluate: 12 − 4 + 3 × 2
Show answer
First, handle multiplication: 3 × 2 = 6. The expression becomes 12 − 4 + 6. Next, work left to right for addition and subtraction: 12 − 4 = 8, then 8 + 6 = 14. The final answer is 14.

Question 3: INTERMEDIATE

Evaluate: 2 × (8 − 3)² ÷ 5 + 1
Show answer
Start with parentheses: 8 − 3 = 5. Then exponents: 5² = 25. Now the expression is 2 × 25 ÷ 5 + 1. Multiply and divide left to right: 2 × 25 = 50, then 50 ÷ 5 = 10. Finally, add: 10 + 1 = 11. The final answer is 11.

Question 4: APPLIED

You buy 3 notebooks at $4 each and 2 pens at $1 each. Your friend gives you a $5 coupon. Write an expression for the total cost and evaluate it using order of operations.
Show answer
The expression is 3 × 4 + 2 × 1 − 5. First, multiply: 3 × 4 = 12 and 2 × 1 = 2. The expression becomes 12 + 2 − 5. Now add and subtract left to right: 12 + 2 = 14, then 14 − 5 = 9. The total cost after the coupon is $9.

Question 5: CRITICAL THINKING

Place one pair of parentheses in the expression 2 + 3 × 4 − 1 so that the result equals 19. Where do the parentheses go, and why does that change the answer?
Show answer
Place the parentheses around 4 − 1 to get 2 + 3 × (4 − 1). Inside the parentheses: 4 − 1 = 3. Then multiply: 3 × 3 = 9. Finally, add: 2 + 9 = 11. That gives 11, not 19. Try (2 + 3) × 4 − 1 instead. Parentheses first: 2 + 3 = 5. Multiply: 5 × 4 = 20. Subtract: 20 − 1 = 19. The parentheses around 2 + 3 force the addition to happen before the multiplication, changing the answer from the default value of 13 to 19. This shows how parentheses can override the normal order.

SUMMARY

Lesson Summary

Summary

The order of operations (remembered by the acronym PEMDAS) is the set of rules that tells you which part of an expression to evaluate first. Start with Parentheses, then Exponents, then Multiplication and Division from left to right, and finally Addition and Subtraction from left to right.

Remember that Multiply/Divide are equal-priority partners, and so are Add/Subtract. Within each pair, you simply move left to right. Following PEMDAS carefully guarantees that every expression gives exactly one correct answer — and these same rules will carry you through algebra, geometry, and beyond.

Varsity Tutors • TACHS Math • Use order of operations to evaluate expressions