Order of Operations Explained: Mastering PEMDAS

Why does the expression 2 plus 3 times 4 equal 14 and not 20? The answer is the order of operations, a universal set of rules that tells everyone the exact sequence in which to carry out a calculation. Without these rules, the same expression could produce different answers depending on who solved it. The popular memory aid PEMDAS captures the order in a single word, and once you understand what it really means, complicated-looking expressions become straightforward and predictable.

What PEMDAS Stands For

PEMDAS is an acronym for the order in which operations should be performed:

• P for Parentheses (and other grouping symbols like brackets)
• E for Exponents (powers and roots)
• M and D for Multiplication and Division
• A and S for Addition and Subtraction

In some countries the same idea is taught as BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). The names differ but the underlying rules are identical. The most important detail, which trips up many students, is that multiplication and division share the same rank, and addition and subtraction share the same rank. Learning these foundations will also make our guide on more guides in algebra much easier to follow.

The One Rule People Forget

It is tempting to read PEMDAS strictly left to right as six separate steps, doing all multiplication before any division. That is incorrect. Multiplication and division are performed together, working from left to right in the order they appear. The same applies to addition and subtraction. So in the expression 12 divided by 3 times 2, you do not multiply first. You move left to right: 12 divided by 3 is 4, then 4 times 2 is 8. If you wrongly multiplied first, you would get 12 divided by 6, which is 2, and that answer is wrong.

A Full Worked Example

Let us evaluate this expression carefully: 6 plus 2 times (5 minus 3) squared divided by 4.

1. Parentheses first. Inside the parentheses, 5 minus 3 equals 2. The expression becomes 6 plus 2 times 2 squared divided by 4.
2. Exponents next. Two squared is 4. Now we have 6 plus 2 times 4 divided by 4.
3. Multiplication and division, left to right. First 2 times 4 is 8, giving 6 plus 8 divided by 4. Then 8 divided by 4 is 2, giving 6 plus 2.
4. Addition and subtraction, left to right. Finally, 6 plus 2 equals 8.

Notice how the answer depends entirely on respecting the order. Each step removes one layer of complexity until only simple arithmetic remains. You can try similar expressions on our practice problems page or check your steps with the free math tools.

Nested Grouping Symbols

Sometimes an expression contains parentheses inside brackets. When this happens, work from the innermost grouping outward. Consider 2 times [3 plus (8 minus 6)]. Start with the innermost parentheses: 8 minus 6 is 2. Now the expression is 2 times [3 plus 2], which is 2 times 5, equal to 10. Always resolve the deepest set of grouping symbols first, then move outward one layer at a time.

How Exponents Fit In

Exponents are evaluated after grouping but before any multiplication, division, addition, or subtraction. Be careful with negative signs. The expression negative 3 squared can be read two ways. If it is written as negative open-parenthesis 3 close-parenthesis squared, the negative is inside, so the answer is 9. But if it is written as negative 3 squared without parentheses around the negative, the convention is to square 3 first and then apply the negative, giving negative 9. To learn more about how powers behave, our overview of exponents in the more guides library goes into depth.

Common Mistakes to Avoid

• Treating multiplication as always before division. They share equal priority; go left to right.
• Ignoring invisible multiplication. A number next to a parenthesis, like 3(4), still means multiply.
• Distributing carelessly. In 2(3 plus 4), simplify inside first to get 2 times 7, or distribute correctly as 6 plus 8. Both give 14.
• Forgetting the left-to-right rule for subtraction. The expression 10 minus 4 minus 3 equals 3, not 9, because you subtract in order.

A Quick Practice Set

Try these and check your reasoning. First, 8 minus 2 times 3 equals 8 minus 6, which is 2. Second, (8 minus 2) times 3 equals 6 times 3, which is 18. Notice how a single pair of parentheses completely changes the result. This is exactly why grouping symbols sit at the top of the order: they let the writer control the sequence.

Frequently Asked Questions

Is PEMDAS the same as BODMAS?

Yes. They use different words but describe the same order of operations. Orders in BODMAS means exponents, and brackets means parentheses.

Do I always multiply before I divide?

No. Multiplication and division have equal priority. Perform them from left to right in the order they appear in the expression.

What if there are no parentheses or exponents?

Then you simply handle multiplication and division left to right, followed by addition and subtraction left to right.

Why do we even need an order of operations?

So that everyone gets the same answer from the same expression. A shared convention removes ambiguity in math, science, and programming.

Once the order of operations becomes second nature, you will read expressions the way a fluent speaker reads a sentence. Work through several examples by hand, then verify your results using our free math tools to lock in the habit.