➕ Algebra 1 · Equations
Order of Operations
Learn the order of operations, why multiplication and division tie, and why left to right matters, with a ladder diagram, memory tricks, games and practice.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- 6.EE.A.1: Write and evaluate numerical expressions involving whole-number exponents.
- 6.EE.A.2c: Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
By the end you can…
- Evaluate expressions with several operations in the agreed order: grouping, exponents, multiplication and division, then addition and subtraction.
- Work multiplication and division (and addition and subtraction) from left to right when they share a level.
- Handle the awkward cases: nested grouping, fraction bars, a minus sign in front of a power, and a negative number substituted for a variable.
Lesson (school style)
An expression like 3 + 4 × 5 could mean two different things: add first and get 35, or multiply first and get 23. To make sure everyone gets the same answer, mathematicians agreed on a ranking called the order of operations: grouping first, then exponents, then multiplication and division, then addition and subtraction. Operations on the same level go from left to right.
Why we need an agreement
Try 3 + 4 × 5. If you work strictly left to right, you get 7 × 5 = 35. If you multiply first, you get 3 + 20 = 23. Both feel natural, so we need a rule that picks one.
The agreed answer is 23. The rule is a convention: an agreement people made so that textbooks, teachers and classmates all match. It is not a law of nature, but nearly all math writing and scientific calculators follow it, so you can rely on it.
The four levels
Work from the top level down:
- Grouping: parentheses ( ), brackets [ ], braces { }, absolute value bars, and a fraction bar (it groups the top and the bottom). Do what is inside first, starting with the innermost group.
- Exponents: evaluate powers such as 42.
- Multiplication and division: these two share one level.
- Addition and subtraction: these two share one level.
Inside a group the same order applies, so a group is like a small problem of its own.
Ties go left to right
Multiplication and division are on the same level. So are addition and subtraction. When two operations from the same level meet, work them from left to right, the way you read.
In 100 ÷ 10 × 5, go left to right: 100 ÷ 10 = 10, then 10 × 5 = 50. (Multiplying first would give 100 ÷ 50 = 2, which is the value of 100 ÷ (10 × 5), a different expression.)
In 50 − 20 + 10, go left to right: 50 − 20 = 30, then 30 + 10 = 40. (Adding first would give 50 − 30 = 20, which is the answer to 50 − (20 + 10).)
This makes sense because subtraction is adding the opposite: 50 − 20 + 10 means 50 + (−20) + 10, and the minus sign stays attached to the 20.
Grouping symbols
Grouping symbols override the normal ranking: what is inside gets finished first, as one package. (2 + 3) × 4 is 5 × 4 = 20, while 2 + 3 × 4 is 2 + 12 = 14. Same numbers, different meaning.
- Nested groups: start at the innermost pair and work outward. In 2 × (3 + (10 − 4)), do 10 − 4 = 6, then 3 + 6 = 9, then 2 × 9 = 18.
- Fraction bar: the top and the bottom are each worked out first, then divided. The fraction with top 20 − 8 and bottom 2 + 4 means (20 − 8) ÷ (2 + 4) = 12 ÷ 6 = 2.
- Absolute value bars: work out what is inside, then take the absolute value: |2 − 9| + 1 = |−7| + 1 = 7 + 1 = 8.
- Hidden multiplication: 3(4 + 2) means 3 × (4 + 2).
- Division by zero: if a group works out to 0 and you divide by it, as in 9 ÷ (4 − 4), the expression is undefined.
Exponents touch only what is next to them
An exponent applies only to the number (or group) right before it. In 2 × 32, only the 3 is squared: 2 × 9 = 18. To square the whole product you need parentheses: (2 × 3)2 = 36.
Negatives need care. In −52 the exponent touches only the 5, so you square 5 first and then apply the minus sign: −(52) = −25. With parentheses, (−5)2 = (−5)(−5) = 25. When you type into a calculator or spreadsheet, use parentheses to show what you mean: (−5)2 for 25, and −(52) for −25. (Some spreadsheets read −52 typed without parentheses as 25, so do not leave it to chance.)
Plugging in a variable
To evaluate an expression such as x2 − 4x + 1 for a value of x, replace each x with the value in parentheses, then follow the order. The parentheses matter most when the value is negative, because they keep the exponent and the minus sign from tangling. The last worked example shows it.
Key words
- Expression
- A math phrase made of numbers, operations and sometimes variables, such as 3 + 4 × 5. It has no equal sign.
- Evaluate
- To find the single number an expression is worth. If there are variables, replace them with the given values first.
- Order of operations
- The agreed ranking that says which operation to do first: grouping, exponents, multiplication and division, then addition and subtraction.
- Grouping symbol
- A symbol that bundles part of an expression so it is worked out first, such as ( ), [ ], { }, absolute value bars or a fraction bar.
- Exponent
- The small raised number on a base. For a positive whole-number exponent, it says how many times the base is used as a factor. In 43 = 4 × 4 × 4, the base is 4, the exponent is 3, and 43 is called a power.
Worked examples
Example 1. Evaluate 9 + 4 × 6 − 8 ÷ 2.
- The multiplication and division level comes before the addition and subtraction level. Work that level left to right: 4 × 6 = 24, then 8 ÷ 2 = 4.
- Rewrite the expression with those results: 9 + 24 − 4.
- Now the addition and subtraction level, left to right: 9 + 24 = 33, then 33 − 4 = 29.
- Check the common slip: working strictly left to right through every symbol gives ((9 + 4) × 6 − 8) ÷ 2 = 35, which breaks the rule.
Answer: Answer: 29
Example 2. Evaluate 48 ÷ 6 × 2, and then evaluate 48 ÷ (6 × 2).
- First expression: division and multiplication share a level, so go left to right: 48 ÷ 6 = 8, then 8 × 2 = 16.
- Second expression: the parentheses go first, so 6 × 2 = 12.
- Then 48 ÷ 12 = 4.
- The same numbers and symbols give different answers because the parentheses change what is being divided by.
Answer: Answer: 16 and 4
Example 3. Evaluate 10 − 2(6 − 3)2.
- Grouping first: 6 − 3 = 3. The expression becomes 10 − 2 × 32, where the 2 multiplies the squared group.
- Exponents next: 32 = 9, giving 10 − 2 × 9. The 2 is not squared, because the exponent touches only the 3.
- Multiplication before subtraction: 2 × 9 = 18, giving 10 − 18.
- Subtract: 10 − 18 = −8.
Answer: Answer: −8
Example 4. Evaluate 90 ÷ [3 + (8 − 5) × 4].
- Find the innermost group: (8 − 5) = 3. The expression is now 90 ÷ [3 + 3 × 4].
- Inside the brackets the usual order still applies, so multiply before adding: 3 × 4 = 12.
- The brackets now hold 3 + 12 = 15, so the expression is 90 ÷ 15.
- Divide: 90 ÷ 15 = 6.
Answer: Answer: 6
Example 5. Evaluate −52 and (−5)2.
- In −52 the exponent touches only the 5, so square the 5 first: 52 = 25.
- Then apply the minus sign: −25.
- In (−5)2 the parentheses put the negative sign inside the square, so it means (−5)(−5).
- A negative times a negative is positive, so (−5)2 = 25.
Answer: Answer: −52 = −25 and (−5)2 = 25
Example 6. Evaluate the fraction whose top is 11 + 2 × 7 and whose bottom is 1 + 4.
- The fraction bar groups the top and the bottom, so treat it as (11 + 2 × 7) ÷ (1 + 4).
- Top: multiply first, 2 × 7 = 14, then add: 11 + 14 = 25.
- Bottom: 1 + 4 = 5.
- Divide the top by the bottom: 25 ÷ 5 = 5.
Answer: Answer: 5
Example 7. Evaluate x2 − 4x + 1 when x = −3.
- Replace every x with (−3), keeping the parentheses: (−3)2 − 4(−3) + 1.
- Exponent first: (−3)2 = 9.
- Multiplication next: 4(−3) = −12. Now the expression is 9 − (−12) + 1.
- Addition and subtraction, left to right: 9 − (−12) = 9 + 12 = 21, then 21 + 1 = 22.
Answer: Answer: 22
In plain English
When a math problem has more than one operation, the answer can change depending on which one you do first. So everyone agrees on one order. That way you, your teacher and most calculators get the same answer.
Here is the order. First, do anything inside parentheses. Second, do powers, like 42. Third, do multiplying and dividing, going left to right. Fourth, do adding and subtracting, going left to right.
Try 3 + 4 × 5. Multiplying comes before adding, so do 4 × 5 = 20 first. Then 3 + 20 = 23. If you just went left to right, you would get 35, and that breaks the agreement.
Multiplying and dividing share a spot, and so do adding and subtracting. When two of them meet, go in reading order, left to right.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Two equally close friends
Two friends who are equally close to you both text you. Neither one outranks the other, so you reply in the order the messages arrived: first come, first served.
How it connects: Multiplication and division are equals, so you go left to right: 30 ÷ 5 × 2 = 6 × 2 = 12, not 30 ÷ 10 = 3. The same goes for addition and subtraction. This tie-breaker works only inside one level: a higher level, such as exponents, still goes before multiplication and division.
Inception: dreams inside dreams
In the movie Inception, a team enters a person’s dream, then a dream inside that dream, and goes deeper still. The deepest dream is nested inside all the others.
How it connects: Parentheses can nest too. In 4 × (1 + (8 − 3)) you can spot the layers from the outside in, but you calculate from the innermost layer outward: 8 − 3 = 5, then 1 + 5 = 6, then 4 × 6 = 24.
Reading a box score
A basketball player makes 7 two-point baskets, 3 three-point baskets and 2 free throws worth 1 point each. Her points are 7 × 2 + 3 × 3 + 2 × 1.
How it connects: Multiplication ranks above addition: first find the points from each kind of shot, then add them: 14 + 9 + 2 = 25. Working strictly left to right through the symbols would give 53, a score no scoreboard would accept.
Three friends, one bill
Three friends each order a 7-dollar burger and a 2-dollar drink. The total is 3 × (7 + 2): find the cost of one friend’s meal first (7 + 2 = 9), then multiply by 3 to get 27 dollars.
How it connects: Parentheses are worked out first, as one package. Without them, 3 × 7 + 2 = 23 would mean three burgers but only one drink, so parentheses can change what an expression means.
Doubling with every octave
Each time a note goes up one octave, its frequency doubles. Starting from 110 hertz and going up three octaves gives 110 × 23 hertz.
How it connects: The exponent touches only the 2 (three doublings), and exponents rank above multiplication: 110 × 23 = 110 × 8 = 880 hertz. Multiplying first would give (110 × 2)3, which is over ten million: a very different expression.
A health bar
In a game your health is 20. You take 6 damage, then you heal 4. That story is 20 − 6 + 4.
How it connects: Subtraction and addition share a level, so go left to right: 20 − 6 = 14, then 14 + 4 = 18. Adding first would give 20 − (6 + 4) = 10, which is a different story (10 damage). The minus sign belongs to the 6, so you can also read it as 20 + (−6) + 4.
First things first
The proverb means you should deal with the most important or most necessary thing before anything else.
How it connects: On the ladder, “first” means highest rank: grouping, then exponents, then multiplication and division, then addition and subtraction. Position on the page only breaks ties between operations of the same rank.
Why exponents go first
Why do exponents get to go before multiplication? Because they have all the power.
How it connects: An exponent is repeated multiplication, so it ranks one level higher: 3 × 24 = 3 × 16 = 48, not 64 = 1296. (A power is the result of using an exponent, which is where the pun comes from.)
Memory help
PEMDAS, with a twist
Please Excuse My Dear Aunt Sally stands for Parentheses (grouping), Exponents, Multiplication, Division, Addition, Subtraction. The twist: M and D tie, and A and S tie. Think of it as PE(MD)(AS). Inside each pair, go left to right, not in the order of the letters.
The ladder rhyme
Groups go first, then powers climb. Times and divide share one line. Plus and minus come at the end. Ties go left to right, my friend.
The four-rung ladder
Picture a ladder with four rungs and step down from the top: grouping, exponents, then multiplication and division, then addition and subtraction. The bottom two rungs are wide enough for two operations each. When two operations stand on the same rung, they take turns from left to right.
Walk through your school
The front door is grouping: you have to go inside first, starting with the innermost room. The gym is exponents (power!). In the cafeteria, multiplication and division share one long table and are served left to right. At the library at the end of the hall, addition and subtraction share the last table, also served left to right.
Rewrite after every step
On paper, change only one thing per line. Underline the operation you are doing, solve it, and rewrite the whole expression with the result in its place. Doing three steps in your head and jumping to the answer is how slips sneak in.
The exponent hugs the number next to it
In −92 the exponent touches only the 9, so the value is −81. If you want the negative squared, add parentheses: (−9)2 = 81.
Study notes
Key ideas
- The order of operations is an agreement: grouping, then exponents, then multiplication and division, then addition and subtraction.
- Multiplication and division share a level. Addition and subtraction share a level. Within a level, work from left to right.
- Grouping symbols include parentheses, brackets, braces, absolute value bars and fraction bars. Do the innermost group first, using the same order inside it.
- An exponent applies only to what it touches: −a2 means −(a2), but (−a)2 means a2.
Steps
- Find the innermost grouping symbol and work it out, using the same order inside.
- Evaluate exponents.
- Do multiplication and division from left to right.
- Do addition and subtraction from left to right.
- Rewrite the whole expression after every step, and check that the answer is reasonable.
Formulas
- The ladder: Grouping, then exponents, then × and ÷ (left to right), then + and − (left to right)
- Minus sign and exponent: −a2 = −(a2), but (−a)2 = a2
- Fraction bar: A fraction bar with top a + b and bottom c + d means (a + b) ÷ (c + d), when c + d ≠ 0
Watch out
- PEMDAS is not a six-step staircase. M and D tie, and A and S tie, so work each pair left to right: 15 − 9 + 2 = 8, not 15 − 11 = 4.
- Going left to right through every symbol is not the rule. Only operations on the same level go left to right.
- The exponent touches only the number right before it, so 2 × 32 = 18, not 36.
- A fraction bar groups its top and its bottom. When you type it on one line, add parentheses.
- When you substitute a negative number for a variable, wrap it in parentheses.
- If a group in a divisor works out to 0, the expression is undefined.
Why it works
- Why have a rule at all? Without one, 3 + 4 × 5 would have two answers. A shared rule lets people (and most calculators and software) agree.
- Why does the ranking look this way? Each level is a shortcut for repeating the level below it: for whole numbers, multiplication is repeated addition and an exponent is repeated multiplication. With these ranks you can write 2x2 + 3x + 1 with no parentheses at all.
Common mistakes
Teach it back
Explain to a friend (or a stuffed animal) how to evaluate 8 + 18 ÷ (5 − 2) × 3 and why each step happens in that order.
Explaining it out loud shows what you really know. Did you…
- I did the parentheses first: 5 − 2 = 3.
- I said that division and multiplication tie, so I went left to right: 18 ÷ 3 = 6, then 6 × 3 = 18.
- I added last and got 8 + 18 = 26.
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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