Why do we need order of operations?

Without the order of operations, different people might solve the same problem in different ways and get different answers. The order of operations is a rule everyone follows so that there is only one correct answer. For example, 2 + 3 \cdot 4 could be 5 \cdot 4 = 20 if you add first, or 2 + 12 = 14 if you multiply first. By following the order of operations (multiply first), everyone gets 14.

Is multiplication always done before division?

No. Multiplication and division have equal priority and are done from left to right. For example, in 12 \div 3 \cdot 2, you do the division first (left to right): 12 \div 3 = 4, then 4 \cdot 2 = 8. If you did multiplication first you would get the wrong answer.

Can you use parentheses to break the order of operations?

Yes. Parentheses have the highest priority in the order of operations, so whatever is inside them is calculated first. For example, normally 2 + 3 \cdot 4 = 2 + 12 = 14, but if you add parentheses like (2 + 3) \cdot 4, you must do the addition first to get 5 \cdot 4 = 20.

Do exponents come before multiplication?

Yes. Exponents are done after parentheses but before any multiplication, division, addition, or subtraction. For example, in 2 \cdot 3^2, you must calculate 3^2 = 9 first to get 2 \cdot 9 = 18. If you multiply first, you would incorrectly get 6^2 = 36.

Order of Operations

Order of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.

\text{P} \to \text{E} \to \text{M},\text{D} \to \text{A},\text{S}

Solve a problem with Order of Operations

Type the problem. The solver will use Order of Operations where Order of Operations is the right tool, and tell you when it is not.

What each symbol means

Symbol	Meaning

When to use it

Use Order of Operations when you have a calculation with multiple operations and need to know the correct order to get the right answer.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Simple expression with addition and multiplication

Problem

Work out

6 + 3 \cdot 2

$6 + 3 \cdot 2$
We have both addition and multiplication, so we need to follow the Order of Operations.
$3 \cdot 2 = 6$
Multiplication comes before addition, so we calculate $3 \cdot 2$ first.
$6 + 6 = 12$
Now add the result to 6.

Answer: $12$

We followed the Order of Operations by doing multiplication before addition. If we had added first, we would get $6 + 3 = 9$ and then $9 \cdot 2 = 18$ , which is wrong.

2. Expression with parentheses, exponents, and all operations

Problem

Work out

2(4-1)^2 + 6 \div 3

$4 - 1 = 3$
Work inside the parentheses first.
$3^2 = 9$
Do the exponent.
$2 \cdot 9 = 18$
Now we have multiplication and division. Do $2 \cdot 9$ and $6 \div 3$ from left to right.
$6 \div 3 = 2$
Continue with division.
$18 + 2 = 20$
Finally, add.

Answer: $20$

This problem needs every part of the Order of Operations. We do parentheses first, then exponents, then multiplication and division together (left to right), and finally addition.

3. Bake sale: calculating total cost

Problem

At a bake sale, boxes of brownies cost $4 each and individual muffins cost $3 each. You buy 2 boxes of brownies and 5 muffins. Work out the total cost using the expression

2 \cdot 4 + 5 \cdot 3

$2 \cdot 4 = 8$
First, multiply: 2 boxes of brownies at $4 each equals $8.
$5 \cdot 3 = 15$
Next multiply: 5 muffins at $3 each equals $15.
$8 + 15 = 23$
Finally add the two costs together.

Answer: $\$23$

The Order of Operations tells us to do multiplication before addition. We must calculate the cost of the brownies and the cost of the muffins separately before adding them together to find the total cost.

Common mistakes

Where Order of Operations usually goes wrong

The mistake	Why it is wrong	The fix
Working left to right and ignoring Order of Operations: $6 + 2 \cdot 3 = 8 \cdot 3 = 24$	Addition comes after multiplication in the Order of Operations, so $2 \cdot 3$ must be calculated first, not $6 + 2$ .	Do multiplication first: $6 + 2 \cdot 3 = 6 + 6 = 12$
Treating division as lower priority than multiplication: $8 \div 2 \cdot 3 = 8 \div 6$	Multiplication and division have the same priority and must be done from left to right, not right to left.	Work left to right with multiplication and division: $8 \div 2 \cdot 3 = 4 \cdot 3 = 12$
Applying the exponent to the whole expression: $2 \cdot 3^2 = 6^2 = 36$	Exponents apply only to the number directly attached to them, not to the entire term. Do exponents first, then multiply.	Calculate the exponent first: $2 \cdot 3^2 = 2 \cdot 9 = 18$

Tips and when to use something else

Remember the order with PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Most people remember this as a catchphrase like 'Please Excuse My Dear Aunt Sally'.
Multiplication and division have equal priority, so work left to right. The same goes for addition and subtraction.
If you are unsure about the order, add parentheses to make it clear what you want to calculate first.
When you want to break up a calculation differently, use the Distributive Property to rewrite it: for example, $3(2 + 5) = 3 \cdot 2 + 3 \cdot 5 = 6 + 15 = 21$ .

Related formulas

Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.Rules of SignsRules of Signs tell you whether the product of two numbers is positive or negative, which you need for solving equations with negative numbers.Absolute ValueAbsolute value measures the distance a number is from zero without regard to direction, so it is always positive or zero.Adding FractionsAdd fractions with different denominators by finding a common denominator, then combining the numerators to get your answer.

Frequently asked questions

Why do we need order of operations?: Without the order of operations, different people might solve the same problem in different ways and get different answers. The order of operations is a rule everyone follows so that there is only one correct answer. For example, $2 + 3 \cdot 4$ could be $5 \cdot 4 = 20$ if you add first, or $2 + 12 = 14$ if you multiply first. By following the order of operations (multiply first), everyone gets 14.
Is multiplication always done before division?: No. Multiplication and division have equal priority and are done from left to right. For example, in $12 \div 3 \cdot 2$ , you do the division first (left to right): $12 \div 3 = 4$ , then $4 \cdot 2 = 8$ . If you did multiplication first you would get the wrong answer.
Can you use parentheses to break the order of operations?: Yes. Parentheses have the highest priority in the order of operations, so whatever is inside them is calculated first. For example, normally $2 + 3 \cdot 4 = 2 + 12 = 14$ , but if you add parentheses like $(2 + 3) \cdot 4$ , you must do the addition first to get $5 \cdot 4 = 20$ .
Do exponents come before multiplication?: Yes. Exponents are done after parentheses but before any multiplication, division, addition, or subtraction. For example, in $2 \cdot 3^2$ , you must calculate $3^2 = 9$ first to get $2 \cdot 9 = 18$ . If you multiply first, you would incorrectly get $6^2 = 36$ .

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Reviewed 2026-09-18