Order of Operations (PEMDAS) Explained

Have you ever solved a math problem, felt completely confident, and then discovered your answer was marked wrong, even though every individual calculation was correct? Nine times out of ten, the culprit is the order of operations. When an expression mixes addition, multiplication, parentheses, and exponents, the order you tackle them in completely changes the result.

In this guide you will learn exactly what the order of operations PEMDAS means, why mathematicians agreed on a single set of rules, and how to apply that order step by step without second-guessing yourself. We will walk through several fully worked examples, point out the traps that catch most students, and show you a reliable checklist you can use on any problem. If you would rather see the steps for a specific problem worked out instantly, our word problem solver can break it down for you in seconds.

By the end, you will be able to look at an expression like \( 8 + 12 \div 4 \times 2 - 3 \) and know precisely which part to evaluate first, second, and last. Let’s build that confidence one rule at a time.

What Is the Order of Operations?

The order of operations is a universal set of rules that tells you the sequence in which to perform calculations in an expression that has more than one operation. Without it, the same expression could give different answers depending on who solves it, which would make math useless for science, engineering, money, and everyday life.

Consider the simple-looking expression \( 2 + 3 \times 4 \). If you work strictly left to right, you would get \( 2 + 3 = 5 \), then \( 5 \times 4 = 20 \). But the agreed-upon order tells us to multiply first: \( 3 \times 4 = 12 \), then \( 2 + 12 = 14 \). The correct answer is 14. Mathematicians worldwide use the same convention so that everyone arrives at that same value.

What Does PEMDAS Stand For?

PEMDAS is the most common memory aid (an acronym) used in the United States to remember the order of operations. Each letter stands for a stage of evaluation:

Letter	Stands for	What it means
P	Parentheses	Solve anything inside grouping symbols first: \( (\;) \), \( [\;] \), \( \{\;\} \)
E	Exponents	Evaluate powers and roots, such as \( 5^2 \) or \( \sqrt{9} \)
M	Multiplication	Multiply and divide together, working left to right
D	Division
A	Addition	Add and subtract together, working left to right
S	Subtraction

Many students remember the acronym with the sentence “Please Excuse My Dear Aunt Sally.” The first letter of each word matches the first letter of each stage.

PEMDAS, BODMAS, and BIDMAS: Same Rules, Different Names

You may have seen other acronyms, especially if you study outside the United States. They all describe the exact same order of operations; only the wording changes.

Acronym	Used in	What the letters mean
PEMDAS	United States	Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
BODMAS	UK, India	Brackets, Orders, Division/Multiplication, Addition/Subtraction
BIDMAS	UK	Brackets, Indices, Division/Multiplication, Addition/Subtraction
GEMS	Some US schools	Grouping, Exponents, Multiplication/Division, Subtraction/Addition

“Brackets” is just another word for parentheses, and “Orders” and “Indices” both mean exponents. Notice that BODMAS lists division before multiplication, while PEMDAS lists multiplication first. This does not mean the two systems disagree. Both still tell you to handle multiplication and division at the same level, left to right.

The PEMDAS Steps in Detail

Let’s slow down and look at each stage so you know exactly what to do.

Step 1: Parentheses and Grouping Symbols

Always start with anything inside grouping symbols. These include round parentheses \( (\;) \), square brackets \( [\;] \), and curly braces \( \{\;\} \). A fraction bar and the inside of a square root also act as grouping symbols. When groups are nested inside one another, work from the innermost group outward.

Step 2: Exponents and Roots

Next, evaluate all exponents (powers) and roots. For example, \( 4^3 = 64 \) and \( \sqrt{25} = 5 \). Exponents tell you how many times to multiply a base by itself, so \( 4^3 = 4 \times 4 \times 4 \).

Step 3: Multiplication and Division (Left to Right)

Now handle multiplication and division together, capturing from left to right. Whichever one you reach first as you read across the expression, you do first. They are equal in rank.

Step 4: Addition and Subtraction (Left to Right)

Finally, do all addition and subtraction, again moving left to right. Like multiplication and division, these two operations share the same priority.

Worked Example 1: A Four-Operation Expression

Let’s evaluate:

$$ 8 + 12 \div 4 \times 2 - 3 $$

There are no parentheses and no exponents, so we jump straight to multiplication and division, working left to right.

Step 1 — Division and multiplication, left to right. Reading across, division comes first:

$$ 12 \div 4 = 3 $$

The expression now reads \( 8 + 3 \times 2 - 3 \). Continuing left to right, do the multiplication:

$$ 3 \times 2 = 6 $$

Now we have \( 8 + 6 - 3 \).

Step 2 — Addition and subtraction, left to right. Add first because it appears first:

$$ 8 + 6 = 14 $$

Then subtract:

$$ 14 - 3 = 11 $$

Notice how the multiplication and division were resolved before either the addition or the subtraction, even though they appear in the middle of the expression. That is PEMDAS in action.

Worked Example 2: Parentheses and Exponents

Now let’s add some grouping and a power:

$$ (5 + 3)^2 \div 4 + 6 \times 2 $$

Step 1 — Parentheses. Solve inside the parentheses first:

$$ 5 + 3 = 8 $$

The expression becomes \( 8^2 \div 4 + 6 \times 2 \).

Step 2 — Exponents. Evaluate the power:

$$ 8^2 = 8 \times 8 = 64 $$

Now we have \( 64 \div 4 + 6 \times 2 \).

Step 3 — Multiplication and division, left to right. Division comes first as we read across:

$$ 64 \div 4 = 16 $$

Then the multiplication:

$$ 6 \times 2 = 12 $$

The expression is now \( 16 + 12 \).

Step 4 — Addition.

$$ 16 + 12 = 28 $$

Worked Example 3: Nested Brackets

This one has grouping symbols inside grouping symbols. We work from the inside out.

$$ 2 \times \left[\, 3 + (8 - 2)^2 \div 4 \,\right] - 5 $$

Step 1 — Innermost parentheses. Start with \( (8 - 2) \):

$$ 8 - 2 = 6 $$

The expression becomes \( 2 \times \left[\, 3 + 6^2 \div 4 \,\right] - 5 \).

Step 2 — Exponent inside the brackets.

$$ 6^2 = 36 $$

Now we have \( 2 \times \left[\, 3 + 36 \div 4 \,\right] - 5 \).

Step 3 — Division inside the brackets. Inside the brackets we still follow PEMDAS, so division comes before addition:

$$ 36 \div 4 = 9 $$

The brackets now contain \( 3 + 9 \).

Step 4 — Addition inside the brackets.

$$ 3 + 9 = 12 $$

The expression simplifies to \( 2 \times 12 - 5 \).

Step 5 — Multiplication, then subtraction.

$$ 2 \times 12 = 24 $$ $$ 24 - 5 = 19 $$

The Mistakes That Trip Students Up

Most order-of-operations errors come from a small handful of misunderstandings. Spot them once and you will avoid them forever.

Mistake 1: Always Multiplying Before Dividing

Because the “M” comes before the “D” in PEMDAS, students often multiply first by reflex. Watch what happens with \( 16 \div 4 \times 2 \):

• Correct (left to right): \( 16 \div 4 = 4 \), then \( 4 \times 2 = 8 \).
• Incorrect (multiply first): \( 4 \times 2 = 8 \), then \( 16 \div 8 = 2 \).

The correct answer is 8, not 2. Division came first because it appears first as you read left to right.

Mistake 2: Always Adding Before Subtracting

The same trap exists for addition and subtraction. Evaluate \( 10 - 4 + 2 \):

• Correct (left to right): \( 10 - 4 = 6 \), then \( 6 + 2 = 8 \).
• Incorrect (add first): \( 4 + 2 = 6 \), then \( 10 - 6 = 4 \).

The correct answer is 8. Subtraction came first here.

Mistake 3: Misreading Negative Signs and Exponents

The expression \( -3^2 \) is a famous trap. By convention, the exponent attaches only to the 3, so:

$$ -3^2 = -(3 \times 3) = -9 $$

If you actually want to square negative three, you must use parentheses:

$$ (-3)^2 = (-3) \times (-3) = 9 $$

The parentheses make the negative sign part of the base, which changes the answer from \( -9 \) to \( 9 \).

Mistake 4: Forgetting That a Fraction Bar Groups

A long fraction bar acts like invisible parentheses around the top and the bottom. Consider:

$$ \frac{12 + 8}{2 + 2} $$

You must evaluate the numerator and denominator separately before dividing: \( 12 + 8 = 20 \) and \( 2 + 2 = 4 \), giving \( 20 \div 4 = 5 \). Writing it on one line as \( 12 + 8 \div 2 + 2 \) would be a different problem entirely, so be careful when you flatten fractions onto a single line.

A Real-World Word Problem Using PEMDAS

The order of operations is not just for textbook drills. It shows up any time you combine quantities. Suppose you are buying school supplies: 3 notebooks that cost 4 dollars each, and 2 pens that cost 2 dollars each. The total cost is captured in a single expression:

$$ 3 \times 4 + 2 \times 2 $$

Following PEMDAS, both multiplications happen before the addition. Notebooks: \( 3 \times 4 = 12 \). Pens: \( 2 \times 2 = 4 \). Then add:

$$ 12 + 4 = 16 $$

If you had ignored the order of operations and added \( 4 + 2 \) before multiplying, you would have gotten a meaningless number. The rules keep the math matched to reality. For more practice turning real situations into math, see our guide on How to Solve Math Word Problems: A Step-by-Step Framework.

Order of Operations with Fractions and Percentages

PEMDAS works exactly the same way when fractions and percentages are involved. The fraction bar groups, and a percent is just a number you eventually convert. For example, to find 20 percent of the sum \( 30 + 20 \), you first resolve the grouped sum, then multiply:

$$ 0.20 \times (30 + 20) = 0.20 \times 50 = 10 $$

If your expression mixes fractions, remember to apply the same four tiers in order. Our companion guides on How to Add, Subtract, Multiply and Divide Fractions and How to Calculate Percentages (with Examples) show the arithmetic in detail once the order is sorted out.

A Quick PEMDAS Checklist

Keep this five-point routine in your head and you will rarely slip:

1. Look it over first. Read the whole expression before touching anything, and notice where the grouping symbols are.
2. Clear the groups. Simplify everything inside parentheses, brackets, and braces, working innermost first.
3. Power up. Evaluate every exponent and root.
4. Multiply and divide left to right.
5. Add and subtract left to right, and write your final answer.

Why the Order of Operations Matters Beyond the Classroom

The order of operations PEMDAS is the quiet backbone of every calculator, spreadsheet formula, and computer program. When you type a formula into a spreadsheet to total a budget, the software follows these exact rules. When a scientist writes an equation, when a banker computes interest, or when an engineer sizes a beam, the same convention guarantees that everyone reads the math the same way.

That is why mastering this skill early pays off for years. It is foundational for algebra, where you will simplify expressions with variables, and essential for evaluating formulas correctly. Get comfortable now, and the harder topics ahead will feel far more approachable.

Conclusion

The order of operations gives every multi-step expression a single, correct answer. Remember the four tiers behind PEMDAS: clear the parentheses, evaluate the exponents, handle multiplication and division left to right, and finish with addition and subtraction left to right. Watch out for the side-by-side traps, respect the fraction bar, and rewrite the expression after each step. With a little practice, expressions that once looked intimidating will feel routine.

Ready to put it into practice? Try a problem of your own with our free word problem solver to see every step worked out, and explore more clear, friendly tutorials over on the Math Solver AI blog. Keep practicing and trusting the order, and it will never let you down.