Order of Operations: Definition, PEMDAS Rules & Examples

Definitionnoun phrase

What is the order of operations in math?

The order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.

In Spanish: orden de las operaciones
Also called: PEMDAS, BODMAS, BIDMAS, GEMDAS
Grade level: 6-8

Grade 5Grade 6Grade 7Grade 8CCSS 5.OA.A.1CCSS 6.EE.A.2.cGEDSATACT

Order of Operations in a picture

One operation at a time, in the agreed order: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right). The green part is what gets simplified in each step.

Formal definition and notation

To evaluate an expression: (1) simplify inside grouping symbols, such as parentheses, brackets, braces, a fraction bar or absolute value bars, working from the innermost group out; (2) evaluate exponents and roots; (3) do all multiplication and division in order from left to right, because the two have equal priority; (4) do all addition and subtraction from left to right, also with equal priority. The order is a convention that everyone agrees on, so that a written expression has exactly one value.

Symbol	Read it as	Meaning
( ) [ ] { }	parentheses, brackets, braces	grouping symbols: work inside them first, innermost first
8 + 42 + 1	8 plus 4, over 2 plus 1	a fraction bar groups the top and the bottom: 12 ÷ 3 = 4
5²	five squared	an exponent, done before any multiplication: 2 × 5² = 50

Key facts

Multiplication does not always come before division: 24 ÷ 4 × 2 = 6 × 2 = 12, working left to right.
Addition does not always come before subtraction: 10 − 4 + 3 = 6 + 3 = 9.
Grouping symbols include brackets, braces, a fraction bar and absolute value bars, not only parentheses.
An exponent applies only to what it touches: −3² = −9, but (−3)² = 9.
Scientific calculators follow the order of operations, so type parentheses where you want something done first.

PEMDAS: the order, step by step

P → E → MD → AS

P: parentheses and other grouping symbols
E: exponents (and roots)
MD: multiplication and division, one tie, left to right
AS: addition and subtraction, one tie, left to right

A memory aid is Please Excuse My Dear Aunt Sally. M is not ahead of D, and A is not ahead of S: in each pair, whichever comes first from the left goes first. BODMAS, BIDMAS and GEMDAS name the same order.

Order of Operations examples, step by step

Example 1: Division and multiplication tie

Warm-up

Evaluate 18 − 6 ÷ 3 × 2.

No parentheses or exponents. Multiplication and division come next, left to right.
Division is first from the left: 6 ÷ 3 = 2, giving 18 − 2 × 2.
Then multiply: 2 × 2 = 4, giving 18 − 4.
Subtract: 18 − 4 = 14.

Answer: The value is 14.

Example 2: Nested grouping symbols

Core

Evaluate 5 × [2 + (9 − 4)²] − 3³.

Innermost parentheses first: 9 − 4 = 5, so the brackets hold 2 + 5².
Exponent inside the brackets: 5² = 25, so the brackets hold 2 + 25 = 27.
Now the expression is 5 × 27 − 3³. Exponent: 3³ = 27.
Multiply: 5 × 27 = 135. Subtract: 135 − 27 = 108.

Answer: The value is 108.

Example 3: A negative sign and an exponent

Test-style

What is the value of −2² + 10 ÷ (1 + 4)?

Parentheses: 1 + 4 = 5.
Exponent: 2² = 4. The minus sign is not inside parentheses, so −2² = −4.
Division: 10 ÷ 5 = 2.
Add: −4 + 2 = −2. The trap answer 6 comes from squaring −2.

Answer: The value is -2.

Common mistakes

✗ Mistake24 ÷ 4 × 2 = 24 ÷ 8 = 3, because M comes before D in PEMDAS.

✓ Instead24 ÷ 4 × 2 = 6 × 2 = 12.

Multiplication and division are done left to right; addition and subtraction are also done left to right. Neither one in a pair outranks the other.

✗ Mistake2 + 3 × 4 = 5 × 4 = 20, working straight from left to right.

✓ Instead2 + 3 × 4 = 2 + 12 = 14.

Multiplication comes before addition, wherever it sits in the expression.

✗ Mistake10 − 4 + 3 = 10 − 7 = 3, adding first.

✓ Instead10 − 4 + 3 = 6 + 3 = 9.

Addition and subtraction tie, so go from left to right. The subtraction comes first here.

✗ Mistake−3² = 9.

✓ Instead−3² = −(3²) = −9. Only (−3)² equals 9.

The exponent applies to the 3 alone; the minus sign is applied after the exponent.

Not to be confused with

Order of Operationsvs.Expression

An expression is the math phrase itself, like 2 + 3 × 4. The order of operations is the rule you follow to find its value, 14.

Order of Operationsvs.Distributive Property

The distributive property is a second, legal path that gives the same value: 4(6 + 2) = 4 × 8 = 32 and also 24 + 8 = 32. It does not break the order of operations.

Quick check: 3 questions

Question 1 of 3

Evaluate 20 − 12 ÷ 4 + 1.

Show answer and explanation

Answer: A. Divide first: 12 ÷ 4 = 3. Then left to right: 20 − 3 = 17 and 17 + 1 = 18. Choice B subtracts before dividing, C adds 3 + 1 before subtracting, and D treats the expression as (20 − 12) ÷ (4 + 1).

Question 2 of 3

Which operation comes first in 7 + 3 × (8 − 2)²?

Show answer and explanation

Answer: C. Parentheses come first: 8 − 2 = 6. The exponent then squares the whole group, 6² = 36, so squaring only the 2 is wrong. The full value is 7 + 3 × 36 = 115.

Question 3 of 3

What is −4² + 3 × 2?

Show answer and explanation

Answer: B. −4² = −16 (the exponent goes on the 4 only), and 3 × 2 = 6, so −16 + 6 = −10. Choice A squares −4, C adds before multiplying, and D does both.

Order of Operations: frequently asked questions

Does multiplication always come before division?

No. Multiplication and division have the same rank, so you do whichever comes first as you read from left to right. In 12 ÷ 3 × 2 you divide first and get 8, not 2. The same goes for addition and subtraction. PEMDAS lists M before D and A before S only because a word has to put its letters in some order.

What is the difference between PEMDAS and BODMAS?

Only the names. PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction) is common in the United States. BODMAS uses brackets and orders (or of), and BIDMAS uses indices, but the order of the steps is the same. GEMDAS says grouping to remind you that brackets, fraction bars and absolute value bars count too.

Why do we need an order of operations?

Without one, a single expression could have several values. Working left to right, 2 + 3 × 4 would be 20; doing the multiplication first gives 14. Agreeing on one order means everyone, including every calculator and spreadsheet, gets 14. It also lets you write expressions with fewer parentheses.

Practice and tools

Effortless Math practicePractice on Effortless Math Order of Operations CalculatorType an expression and see each PEMDAS step Scientific CalculatorCheck your answer; it follows the same order

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 5.OA.A.1, CCSS 6.EE.A.2.c

Cite this entry

Effortless Math Team. “Order of Operations.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/order-of-operations/

Flashcard

TermOrder of OperationsPEMDAS

DefinitionParentheses, exponents, multiplication and division left to right, addition and subtraction left to right. 2 + 3 × 4 = 14, not 20.

Study the full deck

Browse all terms in the Math Dictionary · More algebra terms · Random term

EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 5³, the base is 5 and the exponent is 3, so 5³ = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x² + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3PEMDASPEMDAS is an acronym for remembering the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Multiplication and division share one step and are done from left to right, and so are addition and subtraction. Many students learn it as “Please Excuse My Dear Aunt Sally.”Algebra · Grade 5, Grade 6, Grade 7Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8FormulaA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Problem Solving · Grade 5, Grade 6, Grade 7VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a² and −2a², but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8

Order of Operations

What is the order of operations in math?

Order of Operations in a picture

Formal definition and notation

Key facts

PEMDAS: the order, step by step

Order of Operations examples, step by step

Example 1: Division and multiplication tie

Example 2: Nested grouping symbols

Example 3: A negative sign and an exponent

Common mistakes

Not to be confused with

Quick check: 3 questions

Order of Operations: frequently asked questions

Does multiplication always come before division?

What is the difference between PEMDAS and BODMAS?

Why do we need an order of operations?

Related terms

Practice and tools

Sources

Flashcard

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