Last Updated: March 30, 2026
Learn the order of operations in math with a complete, student-friendly guide to PEMDAS, BODMAS, BEDMAS, and GEMDAS. This page explains the rule, the meaning behind each letter, every major formula, common mistakes, and step-by-step solutions you can actually follow.
Visual quick-reference chart for PEMDAS and BODMAS. Use it near the top of the article so students instantly understand the sequence before reading the detailed explanations.
The order of operations is the universal rule that tells you which calculation to do first when a math expression has more than one operation. In the United States, students usually memorize the rule as PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. In the UK, India, Australia, and many other regions, the same idea is taught as BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
The most important thing students miss is this: multiplication and division are equal in priority, so you work left to right. The same is true for addition and subtraction. That one idea explains why so many “viral” math problems confuse people online.
Best way to remember it:
Grouping symbols → Powers/Orders → Multiply & Divide left to right → Add & Subtract left to right
Usually used in the United States
Memory aid: “Please Excuse My Dear Aunt Sally.”
Common in the UK, India, Australia, and beyond
Different words, same mathematical logic, same final answers.
PEMDAS does not mean “always multiply before divide.” It means multiplication and division belong to the same level. So you solve them from left to right. The exact same thing is true for addition and subtraction. Once students understand that PEMDAS is really P → E → (MD) → (AS), the whole topic becomes much easier.
❌ Wrong shortcut: “M comes before D, so
always multiply first.”
✅ Correct rule: Do multiplication or division in the order they
appear.
❌ Wrong shortcut: “A comes before S, so always add first.”
✅ Correct
rule: Do addition or subtraction in the order they appear.
Students often search for PEMDAS meaning, BODMAS meaning, BODMAS full form, or even define PEMDAS. All of these acronyms point to the same mathematical order. The only real difference is the vocabulary.
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Brackets, Orders, Division, Multiplication, Addition, Subtraction.
Brackets, Exponents, Division, Multiplication, Addition, Subtraction. This is another classroom variation.
Grouping, Exponents, Multiplication, Division, Addition, Subtraction. “Grouping” is broader because it includes fraction bars, radicals, and absolute value bars too.
So when a student asks, “Is BODMAS or PEMDAS correct?” the best answer is: both are correct. They are just different memory aids for the same mathematical order of operations.
Solve anything inside parentheses \( ( ) \), brackets \( [ ] \), braces \( \{ \} \), fraction bars, radical bars, or absolute value bars. Work from the innermost grouping symbol outward.
Example: \( (3 + 5) \times 2 = 8 \times 2 = 16 \)
Then simplify powers, indices, roots, and other exponent-style expressions. In BODMAS, this stage is called Orders. In PEMDAS, it is called Exponents.
Example: \( 3^2 + 4 = 9 + 4 = 13 \)
At this level, multiplication and division are tied. Start at the left and move right, simplifying one operation at a time. That is why \( 18 \div 9 \times 2 \) becomes \( 2 \times 2 = 4 \), not 1.
Example: \( 12 \div 4 \times 3 = 3 \times 3 = 9 \)
Finally, handle addition and subtraction in the order they appear. Think of subtraction as adding a negative if that helps: \( 10 - 3 + 5 = 10 + (-3) + 5 \).
Example: \( 10 - 3 + 5 = 7 + 5 = 12 \)
Students often search for every single formula connected to PEMDAS or BODMAS. Technically, PEMDAS itself is not one formula. It is a procedure that tells you how to simplify many different formulas and expressions. Still, there are essential rules you should know because they show up constantly in order of operations math.
\( a^m \times a^n = a^{m+n} \)
\( \frac{a^m}{a^n} = a^{m-n} \), when \( a \neq 0 \)
\( (a^m)^n = a^{mn} \)
\( (ab)^n = a^n b^n \)
\( a^0 = 1 \), when \( a \neq 0 \)
\( \sqrt{ab} = \sqrt a \sqrt b \), for nonnegative values
\( \sqrt{a^2} = |a| \)
\( \sqrt[n]{a^m} = a^{m/n} \), when defined
Remember: the radical bar is a grouping symbol.
\( \frac{a+b}{c} \neq a + \frac{b}{c} \) unless you split carefully
\( \frac{ab}{c} = a \times \frac{b}{c} \)
A fraction bar groups the entire numerator and denominator.
\( a \div b = a \times \frac{1}{b} \), when \( b \neq 0 \)
\( (-a)^2 = a^2 \)
\( -a^2 = -(a^2) \)
\( (-a)^3 = -a^3 \)
\( a - b = a + (-b) \)
Always watch the parentheses when negatives are involved.
Grouping symbols → Orders/Exponents → Multiplication & Division left to right → Addition & Subtraction left to right
When teachers explain order and operations, they often begin with parentheses. That is useful, but it is not complete. In real expressions, grouping can appear in many forms:
That is why some teachers prefer GEMDAS or Grouping instead of Parentheses. It reminds students that the first stage includes every symbol that encloses a complete subexpression. For example, in \( \frac{3 + 9}{2 + 4} \), you do not just divide immediately. You first simplify the numerator and denominator because the fraction bar groups both parts. The same logic applies to \( \sqrt{1 + 8} \): the radical sign tells you to simplify what is underneath before taking the square root.
Solve: \( 6 \times 4 + 8 \div 2 \)
Step 1: Multiplication and division first, left to right
\( 6 \times 4 = 24 \)
Now: \( 24 + 8 \div 2 \)
\( 8 \div 2 = 4 \)
Now: \( 24 + 4 \)
Step 2: Add
\( 24 + 4 = 28 \)
✓ Answer: 28
Solve: \( (8 + 5) - 3 \times 2^2 \)
Step 1: Parentheses
\( 8 + 5 = 13 \)
Now: \( 13 - 3 \times 2^2 \)
Step 2: Exponent
\( 2^2 = 4 \)
Now: \( 13 - 3 \times 4 \)
Step 3: Multiplication
\( 3 \times 4 = 12 \)
Now: \( 13 - 12 \)
Step 4: Subtraction
\( 13 - 12 = 1 \)
✓ Answer: 1
Solve: \( 18 \div 9 \times 2 \)
This is where students often lose marks.
Step 1: Work left to right
\( 18 \div 9 = 2 \)
Now: \( 2 \times 2 \)
Step 2: Multiply
\( 2 \times 2 = 4 \)
✓ Answer: 4
❌ Wrong method: \( 18 \div (9 \times 2) = 1 \). That wrongly changes the original expression.
Solve: \( \frac{6 + 2}{4 - 2} + 3 \)
Step 1: Simplify the numerator and denominator
Numerator: \( 6 + 2 = 8 \)
Denominator: \( 4 - 2 = 2 \)
Now: \( \frac{8}{2} + 3 \)
Step 2: Divide
\( \frac{8}{2} = 4 \)
Now: \( 4 + 3 \)
Step 3: Add
\( 4 + 3 = 7 \)
✓ Answer: 7
Compare: \( (-3)^2 + 1 \) and \( -3^2 + 1 \)
For \( (-3)^2 + 1 \):
\( (-3)^2 = 9 \)
\( 9 + 1 = 10 \)
For \( -3^2 + 1 \):
Exponent first: \( 3^2 = 9 \)
Apply the negative: \( -9 + 1 = -8 \)
The parentheses change the base. That is why these two expressions are not the same.
Solve: \( 100 \div 5^2 - 3 \times (4 + 2) + 15 \)
Step 1: Parentheses
\( 4 + 2 = 6 \)
Now: \( 100 \div 5^2 - 3 \times 6 + 15 \)
Step 2: Exponent
\( 5^2 = 25 \)
Now: \( 100 \div 25 - 18 + 15 \)
Step 3: Division and multiplication
\( 100 \div 25 = 4 \)
\( 3 \times 6 = 18 \)
Now: \( 4 - 18 + 15 \)
Step 4: Left to right with subtraction and addition
\( 4 - 18 = -14 \)
\( -14 + 15 = 1 \)
✓ Answer: 1
This expression appears constantly in searches related to order of operations solver, PEMDAS calculator, and using order of operations calculator. Here is the standard school-math interpretation:
Problem: \( 8 \div 2(2 + 2) \)
Step 1: Parentheses → \( 2 + 2 = 4 \)
Now: \( 8 \div 2 \times 4 \)
Step 2: Multiplication and division left to right → \( 8 \div 2 = 4 \)
Step 3: \( 4 \times 4 = 16 \)
✓ Standard classroom answer: 16
Why do people argue for 1? Because in some contexts, adjacent multiplication like \( 2(4) \) is treated visually as a tighter unit. But in standard classroom simplification rules, especially for typed expressions, multiplication and division remain at the same level and should be evaluated left to right. The safest way to avoid confusion is to write expressions with clear parentheses.
A lot of students are comfortable with simple whole-number expressions, but then get stuck when the page includes fractions, decimals, or algebraic terms. The good news is that the rule does not change. The same math order of calculation applies whether the expression is basic arithmetic or full algebra.
Treat the fraction bar as a grouping symbol. Simplify the numerator completely, simplify the denominator completely, then divide. For example, \( \frac{2 + 10}{3 + 3} = \frac{12}{6} = 2 \).
Decimals do not change the order. In \( 3.5 + 2 \times 4 \), you still multiply first, then add. So the answer is \( 3.5 + 8 = 11.5 \).
When you simplify expressions such as \( 2x + 3(x - 1)^2 \), you still begin with parentheses, then exponents, then multiplication, then addition. This is one reason order of operations is foundational to algebra.
Implied multiplication means multiplication without a visible \( \times \) sign, such as \( 3a \) or \( 2(x+5) \). It is still multiplication. If a bracket sits next to a number or variable, multiply after simplifying the bracketed expression.
Correct view: it has four levels: P → E → (MD) → (AS).
A fraction groups the whole numerator and denominator. Never simplify only one small part unless the algebra actually allows it.
There is a huge difference between \( (-2)^4 \) and \( -2^4 \). Parentheses decide what the exponent applies to.
Only exponents are evaluated in their natural structure. For MD and AS, you must follow the line from left to right.
A scientific calculator usually follows PEMDAS/BODMAS, but if you enter an expression incorrectly, the calculator will still return the wrong result for the problem you intended to solve.
Do not try to perform two or three different types of operations in your head at once. Rewrite after each simplification.
This helps you see the structure immediately and prevents you from rushing into multiplication or addition too early.
This makes the left-to-right rule much easier to understand in longer expressions.
Clear notation prevents arguments, especially with division and implied multiplication.
If you are solving an algebraic expression, plug values back in or compare with a calculator after simplifying.
Use problems with brackets, exponents, fractions, radicals, and variables so the rule becomes flexible, not memorized in only one form.
Some students ask why this topic matters outside class. The truth is that the order of operations is used whenever a formula contains more than one step. That includes:
In other words, operation for math is not a tiny school rule. It is the grammar that keeps quantitative reasoning consistent.
Try these on your own, then click to reveal the solution.
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Mastering the order of operations is just the first step. Once you are comfortable with PEMDAS and BODMAS, you will regularly apply these rules to more advanced math topics. Connect these concepts to other nearby NUM8ERS math resources:
Students often have questions about the precise definitions of terms like pemdas, bodmas, and the order of operations. Here is the simplest explanation to build your foundation:
PEMDAS meaning: PEMDAS is the rule used to solve expressions with more than one operation. BODMAS is exactly the same rule translated with slightly different words. Whether you use the PEMDAS rule or the BODMAS rule, the order of operations tells you to simplify grouped parts first, exponents second, multiplication and division third from left to right, and addition and subtraction last from left to right. This is the entire framework for mathematical calculations.
A visual chart can initially help you remember the concept, but true mastery requires regular application. You will often need to remember whether multiplication comes before division (they are tied!), whether addition comes before subtraction (also tied!), how fraction bars behave, what implied multiplication means, and how to properly evaluate expressions containing exponents or brackets.
Always remember this key principle: PEMDAS is a guide to structure, not a race to the biggest-looking number. You should never choose steps just because a specific number is easy to calculate or because multiplication "looks more important." You must choose your next calculation solely because the structural notation tells you what to simplify first.
If you are teaching this topic at home or in a classroom, one of the best methods is to separate memorization from reasoning. The acronym helps students remember the sequence, but genuine mastery comes from seeing why grouped expressions must be simplified first and why tied operations are handled left to right.
A useful routine is:
Students who keep making mistakes with order of expressions usually need more practice rewriting, not just more reminders about the acronym. In many cases, their real problem is skipping written steps or not seeing that fraction bars and radicals are grouping symbols.
For assessment design, include a mix of routine expressions and deliberately tricky ones. That reveals whether a student genuinely understands the rule or is simply repeating a memory phrase without applying it correctly.
Many students search for an order of operations worksheet, an order of operations worksheet pdf, or even phrases like homework 4 order of operations answers because they need a repeatable method, not just a one-time explanation. A good worksheet method should be almost mechanical. The student should know exactly what to look for on every line.
The strongest routine is:
That may sound slow, but it is exactly what prevents most errors. The majority of wrong answers do not come from “not knowing PEMDAS.” They come from skipping the rewrite step. Students mentally change the structure of the expression, then continue from a line that no longer matches the original problem. Written work prevents that.
Teachers can also organize worksheet sets by difficulty:
If you build practice this way, the learner sees that the order operation in math is always the same, even when the notation grows more complex. That is the real goal of worksheet design: not endless repetition of one simple pattern, but transfer of the rule into new situations.
For homework checking, encourage students to verify one step at a time rather than only checking the final answer. If the final answer is wrong but the structure was correct for the first three steps, the feedback becomes far more useful. It tells the student exactly where the reasoning slipped.
Solve: \( 50 + 20\% \times 200 \)
Convert the percent or think of it as \( 0.20 \). Then multiply first: \( 0.20 \times 200 = 40 \). Now add: \( 50 + 40 = 90 \).
Solve: \( \frac{3 + 9}{2(1 + 2)} \)
Top: \( 3 + 9 = 12 \). Bottom parentheses: \( 1 + 2 = 3 \). Bottom becomes \( 2 \times 3 = 6 \). Then divide: \( \frac{12}{6} = 2 \).
Evaluate \( 2x^2 + 3(x - 1) \) when \( x = 4 \).
Substitute first: \( 2(4)^2 + 3(4 - 1) \). Exponent: \( 4^2 = 16 \). Parentheses: \( 4 - 1 = 3 \). Now multiply: \( 2 \times 16 = 32 \) and \( 3 \times 3 = 9 \). Finally add: \( 32 + 9 = 41 \).
Solve: \( \sqrt{25 - 9} + 2 \times 3 \)
Grouping under the radical: \( 25 - 9 = 16 \). Radical: \( \sqrt{16} = 4 \). Multiplication: \( 2 \times 3 = 6 \). Addition: \( 4 + 6 = 10 \).
Solve: \( |4 - 10| + 3^2 \)
Absolute value grouping first: \( 4 - 10 = -6 \), so \( |-6| = 6 \). Exponent: \( 3^2 = 9 \). Then add: \( 6 + 9 = 15 \).
For quizzes and exams, the biggest time-saver is not speed. It is structure. Students who write the next full line after each step make fewer sign mistakes, fewer exponent mistakes, and fewer fraction mistakes. That is true in arithmetic, algebra, and even advanced topics where formulas get long.
Here are strong exam habits:
Students also benefit from learning the language around the topic. Phrases like order of ops, order of calculation, math the order of operations, and order of expressions all point to the same skill. Seeing the different labels reduces panic when a worksheet or exam uses unfamiliar wording.
Finally, remember that good notation is part of good mathematics. A student who understands PEMDAS but writes carelessly can still lose marks. Clear spacing, visible parentheses, and one step per line are not cosmetic. They are part of correct reasoning.
Students often struggle with the order of operations because the vocabulary changes from one class to another. This glossary helps translate the most common terms into plain language.
A collection of numbers, variables, and operations without an equals sign. Example: \( 4 + 3 \times 2 \).
A calculation such as addition, subtraction, multiplication, division, or exponentiation.
Any symbol that encloses a subexpression and tells you to simplify it first, including parentheses, brackets, braces, fraction bars, radicals, and absolute value bars.
These terms all refer to repeated multiplication or related notation, such as \( x^3 \) or \( a^n \).
Multiplication written without a symbol, such as \( 2x \) or \( 3(x+1) \).
The rule used when two operations share the same level, especially multiplication/division and addition/subtraction.
If students can explain these words out loud, they usually perform better on problems. Vocabulary is not separate from reasoning; it supports reasoning. A learner who knows what a grouping symbol is will notice fraction bars and radicals more quickly. A learner who understands implied multiplication is less likely to make mistakes with expressions like \( 2(x+3) \) or \( 8 \div 2(2+2) \).
That is also why strong pages on this topic should define terms clearly, not just throw a chart at the reader. The chart is useful, but definitions, examples, and careful wording are what turn a reference page into a real learning resource.
First, PEMDAS is not a trick question topic. It becomes confusing only when students rush and stop respecting the original structure of the expression. Slow, clear work usually beats fast mental work.
Second, notation matters. A radical bar, a fraction bar, a negative sign, or a missing set of parentheses can completely change an answer. Strong students learn to read the symbols before they start calculating.
Third, practice should mix formats. If learners only see one style of worksheet, they may think they understand the topic when they really understand only one presentation of it. Use whole numbers, decimals, fractions, variables, radicals, and implied multiplication so the skill becomes durable.
| Order | Operation | Examples | How to work |
|---|---|---|---|
| 1st | Grouping symbols | \( ( ) \), \( [ ] \), \( \{ \} \), fraction bar, radical, absolute value | Innermost first |
| 2nd | Orders / Exponents | \( x^2, x^n, \sqrt x \) | Simplify powers and roots |
| 3rd | Multiplication & Division | \( \times, \div, / \) | Left to right |
| 4th | Addition & Subtraction | \( +, - \) | Left to right |
Grouping symbols → Orders/Exponents → Multiply/Divide (Left to Right) → Add/Subtract (Left to Right)
If you remember only one thing from this guide, remember that PEMDAS and BODMAS are not two competing systems. They are two names for the same mathematical order of operations. The true secret is understanding the tied levels: MD together and AS together.
PEMDAS is a memory aid for the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It helps students solve mixed expressions consistently.
BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. It teaches the same rule as PEMDAS using different terminology.
Yes. Brackets match parentheses, and orders match exponents. The working rule is identical.
No. Multiplication and division share the same level. Work from left to right.
No. Addition and subtraction are also tied. Work from left to right.
Orders refers to powers, exponents, indices, and roots, such as \( 2^3 \) or \( \sqrt{16} \).
BEDMAS is another memory aid: Brackets, Exponents, Division, Multiplication, Addition, and Subtraction. Like PEMDAS and BODMAS, it teaches the same structure.
GEMDAS starts with Grouping instead of Parentheses. This is often more precise because it includes fraction bars, radicals, and absolute value bars too.
Treat the fraction bar as a grouping symbol. Simplify the numerator, simplify the denominator, then divide.
Use the same steps. Parentheses and exponents still come before multiplication and addition, even when variables are present.
Yes, many calculators evaluate expressions using operator precedence. But students should still know the rule so they can enter expressions correctly and spot mistakes.
It is commonly introduced in upper elementary or middle school, then reinforced through algebra and advanced math.
Without it, the same expression could produce different answers for different people. The rule standardizes all arithmetic and algebraic calculation.
It is the exact sequence used to evaluate expressions: grouping, exponents, multiplication/division, and addition/subtraction.
Start with simple examples, emphasize grouping first, and teach PEMDAS as four levels instead of six. Then practice with short expressions before moving to exponents and fractions.
