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Everyone who's taken a math class in the US has heard the acronym "PEMDAS" before. But what does it mean exactly? Here, we will explain in detail the PEMDAS meaning and how it's used before giving you some sample PEMDAS problems so you can practice what you've learned.

PEMDAS Meaning: What Does It Stand For?

PEMDAS is an acronym meant to help you remember the order of operations used to solve math problems. It's typically pronounced "pem-dass," "pem-dozz," or "pem-doss."

Here's what each letter in PEMDAS stands for:

• P arentheses
• M ultiplication and D ivision
• A ddition and S ubtraction

The order of letters shows you the order you must solve different parts of a math problem , with expressions in parentheses coming first and addition and subtraction coming last.

Many students use this mnemonic device to help them remember each letter: P lease E xcuse M y D ear A unt S ally .

In the United Kingdom and other countries, students typically learn PEMDAS as BODMAS . The BODMAS meaning is the same as the PEMDAS meaning — it just uses a couple different words. In this acronym, the B stands for "brackets" (what we in the US call parentheses) and the O stands for "orders" (or exponents). Now, how exactly do you use the PEMDAS rule? Let's take a look.

How Do You Use PEMDAS?

PEMDAS is an acronym used to remind people of the order of operations.

This means that you don't just solve math problems from left to right; rather, you solve them in a predetermined order that's given to you via the acronym PEMDAS . In other words, you'll start by simplifying any expressions in parentheses before simplifying any exponents and moving on to multiplication, etc.

But there's more to it than this. Here's exactly what PEMDAS means for solving math problems:

• Parentheses: Anything in parentheses must be simplified first
• Exponents: Anything with an exponent (or square root) must be simplified after everything in parentheses has been simplified
• Multiplication and Division: Once parentheses and exponents have been dealt with, solve any multiplication and division from left to right
• Addition and Subtraction: Once parentheses, exponents, multiplication, and division have been dealt with, solve any addition and subtraction from left to right

If any of these elements are missing (e.g., you have a math problem without exponents), you can simply skip that step and move on to the next one.

Now, let's look at a sample problem to help you understand the PEMDAS rule better:

4 (5 − 3)² − 10 ÷ 5 + 8

You might be tempted to solve this math problem left to right, but that would result in the wrong answer! So, instead, let's use PEMDAS to help us approach it the correct way.

We know that parentheses must be dealt with first. This problem has one set of parentheses: (5 − 3). Simplifying this gives us 2 , so now our equation looks like this:

4 (2)² − 10 ÷ 5 + 8

The next part of PEMDAS is exponents (and square roots). There is one exponent in this problem that squares the number 2 (i.e., what we found by simplifying the expression in the parentheses).

This gives us 2 × 2 = 4. So now our equation looks like this:

4 (4) − 10 ÷ 5 + 8    OR    4 × 4 − 10 ÷ 5 + 8

Next up is multiplication and division from left to right . Our problem contains both multiplication and division, which we'll solve from left to right (so first 4 × 4 and then 10 ÷ 5). This simplifies our equation as follows:

Finally, all we need to do now is solve the remaining addition and subtraction from left to right :

The final answer is 22. Don't believe me? Insert the whole equation into your calculator (written exactly as it is above) and you'll get the same result!

Sample Math Problems Using PEMDAS + Answers

See whether you can solve the following four problems correctly using the PEMDAS rule. We'll go over the answers after.

Sample PEMDAS Problems

11 − 8 + 5 × 6

8 ÷ 2 (2 + 2)

7 × 4 − 10 (5 − 3) ÷ 2²

√25 (4 + 2)² − 18 ÷ 3 (3 − 1) + 2³

Answer Explanations

Here, we go over each problem above and how you can use PEMDAS to get the correct answer.

#1 Answer Explanation

This math problem is a fairly straightforward example of PEMDAS that uses addition, subtraction, and multiplication only , so no having to worry about parentheses or exponents here.

We know that multiplication comes before addition and subtraction , so you'll need to start by multiplying 5 by 6 to get 30:

11 − 8 + 30

Now, we can simply work left to right on the addition and subtraction:

11 − 8 + 30 3 + 30 = 33

This brings us to the correct answer, which is 33 .

#2 Answer Explanation

If this math problem looks familiar to you, that's probably because it went viral in August 2019 due to its ambiguous setup . Many people argued over whether the correct answer was 1 or 16, but as we all know, with math there's (almost always!) only one truly correct answer.

So which is it: 1 or 16?

Let's see how PEMDAS can give us the right answer. This problem has parentheses, division, and multiplication. So we'll start by simplifying the expression in the parentheses, per PEMDAS:

While most people online agreed up until this point, many disagreed on what to do next: do you multiply 2 by 4, or divide 8 by 2?

PEMDAS can answer this question: when it comes to multiplication and division, you always work left to right. This means that you would indeed divide 8 by 2 before multiplying by 4.

It might help to look at the problem this way instead, since people tend to get tripped up on the parentheses (remember that anything next to a parenthesis is being multiplied by whatever is in the parentheses):

Now, we just solve the equation from left to right:

8 ÷ 2 × 4 4 × 4 = 16

The correct answer is 16. Anyone who argues it's 1 is definitely wrong — and clearly isn't using PEMDAS correctly!

#3 Answer Explanation

Things start to get a bit trickier now.

This math problem has parentheses, an exponent, multiplication, division, and subtraction. But don't get overwhelmed — let's work through the equation, one step at a time.

First, per the PEMDAS rule, we must simplify what's in the parentheses :

7 × 4 − 10 (2) ÷ 2²

Easy peasy, right? Next, let's simplify the exponent :

7 × 4 − 10 (2) ÷ 4

All that's left now is multiplication, division, and subtraction. Remember that with multiplication and division, we simply work from left to right:

7 × 4 − 10 (2) ÷ 4 28 − 10 (2) ÷ 4 28 − 20 ÷ 4 28 − 5

Once you've multiplied and divided, you just need to do the subtraction to solve it:

28 − 5 = 23

This gives us the correct answer of 23 .

#4 Answer Explanation

This problem might look scary, but I promise it's not! As you long as you approach it one step at a time using the PEMDAS rule , you'll be able to solve it in no time.

Right away we can see that this problem contains all components of PEMDAS : parentheses (two sets), exponents (two and a square root), multiplication, division, addition, and subtraction. But it's really no different from any other math problem we've done.

First, we must simplify what's in the two sets of parentheses:

√25 (6)² − 18 ÷ 3 (2) + 2³

Next, we must simplify all the exponents — this includes square roots, too :

5 (36) − 18 ÷ 3 (2) + 8

Now, we must do the multiplication and division from left to right:

5 (36) − 18 ÷ 3 (2) + 8 180 − 18 ÷ 3 (2) + 8 180 − 6 (2) + 8 180 − 12 + 8

Finally, we solve the remaining addition and subtraction from left to right:

180 − 12 + 8 168 + 8 = 176

This leads us to the correct answer of 176 .

What's Next?

Another math acronym you should know is SOHCAHTOA. Our expert guide tells you what the acronym SOHCAHTOAH means and how you can use it to solve problems involving triangles .

Studying for the SAT or ACT Math section? Then you'll definitely want to check out our ultimate SAT Math guide / ACT Math guide , which gives you tons of tips and strategies for this tricky section.

Interested in really big numbers? Learn what a googol and googolplex are , as well as why it's impossible to write one of these numbers out.

Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.

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The Order of Operations: PEMDAS

PEMDAS Worked Examples Multiplication by Juxtaposition Fractions & Grouping

If you are asked to simplify something like " 4 + 2×3 ", the question that naturally arises is "Which way do I do this? Because there are two options!" I could add first:

4 + 2×3 = (4 + 2)×3 = 6×3 = 18

...or I could multiply first:

4 + 2×3 = 4 + (2×3) = 4 + 6 = 10

Which answer is the right one?

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The Order of Operations

It seems as though the answer depends on which way you look at the problem. But we can't have this kind of flexibility in mathematics; math won't work if you can't be sure of the answer, or if the exact same expression can be calculated so that you can arrive at two or more different answers. Mathematics doesn't do "loosey-goosey".

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To fix this "it depends on how you look at it" issue, mathematicians codified an ordering to the arithmetical operations of addition, subtraction, multiplication, division, repeated multiplication (that is, exponentiation), and grouping (that is, parentheticals). This codification of which comes before what is called "the order of operations".

What is the order of operations?

The order of operations is a listing of the basic mathematical operations according to precedence (that is, according to which operation is applied first, then which is applied second, etc). The ordering says that anything inside a parenthetical is simplified first; then exponents are applied; then multiplication and division are applied (going from left to right), and finally addition and subtraction are applied (again, going from left to right).

The order of operations was established informally at least as far back as the 1500s; by "informally", I mean that mathematicians just sort of agreed that the ordering made sense. The order of operations does not seem to have been formalized until the early 1900s. (For more on the history of the formalization of the order of operations, try TheMathDoctors .)

How can I remember the order of operations?

A common technique for remembering the order of operations is the abbreviation (or, more properly, the acronym) "PEMDAS", which has been turned into the mnemonic phrase "Please Excuse My Dear Aunt Sally". This phrase stands for, and helps one remember the order of:

• Parentheses,
• Multiplication and Division, and
• Addition and Subtraction

This listing tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and multiplication and division outrank addition and subtraction (which are together on the bottom rank). In other words, the precedence is:

• Parentheses (simplify inside 'em)
• Exponents (apply them)
• Multiplication and Division (going from left to right)
• Addition and Subtraction (going from left to right)

When you have a bunch of operations of the same rank, you just operate from left to right. For instance:

15 ÷ 3 × 4

...is not 15 ÷ (3 × 4) = 15 ÷ 12 , but is actually:

(15 ÷ 3) × 4 = 5 × 4 = 20

...because, going from left to right, you get to the division sign first.

If you're not sure of this, test it in your calculator, which has been programmed with the Order-of-Operations hierarchy. For instance, typesetting the above expression into a graphing calculator, you will get:

Using the above hierarchy, we see that, in the " 4 + 2 × 3 " question at the beginning of this article, Choice 2 was the correct answer, because we have to do the multiplication before we do the addition.

What is BODMAS? or BEDMAS? or BIMDAS

Speakers of British English often instead use the acronym "BODMAS", rather than "PEMDAS". BODMAS stands for "Brackets, Orders, Division and Multiplication, and Addition and Subtraction". Since "brackets" are grouping symbols like parentheses, and since "orders" is another word for exponents, the two acronyms mean the same thing. (The I in BIMDAS stands for "indices", which is another word for exponents.) Also, note that the "M" and the "D" are reversed in the British-English version; this confirms that multiplication and division are at the same "rank" or "level".

Canadian English-speakers split the difference, using BEDMAS.

Why does "order" mean "exponent"?

In case you're wondering why British-English uses the word "orders" to stand for "exponents", it appears that nobody knows . The Wikipedia source for this equivalence is — I'm not making this up — the page you're reading right now. I can find no formal etymology for this usage. The canonical source for math-word origins only mentions order in the sense of the degree; its closest approach to orders as exponents is "parodick degrees", which refers to a polynomial having its terms in decreasing order. So your guess as to "why 'orders'?"is probably as good as anybody else's.

The order of operations was settled upon in order to prevent miscommunication, but PEMDAS can generate its own confusion. Many students using PEMDAS think that multiplication *must* be done before division; students using BODMAS think that addition *must* be done before subtraction. But multiplication and divison are on the same level, because division is just multiplication by the reciprocal; and addition and subtaction are on the same level, because subtraction is just addition of a negative. (The acronym could maybe be restated as BOPS , meaning "brackets, orders, products, and sums".)

Probably the best way to explain this is to work through some examples:

• Simplify 4 + 3 2 .

I need to simplify the term with the exponent before trying to add it to the 4:

• Simplify 4 + (2 + 1) 2 .

I have to simplify inside the parentheses before I can take the exponent through. Only then can I deal with the addition of the 4 .

4 + (2 + 1) 2

= 4 + (3) 2

• Simplify 4 + [−1(−2 − 1)] 2 .

I shouldn't try to do these nested parentheses from left to right; attempting to simplify that way is way too error-prone. Instead, I'll try to work from the inside out.

First I'll simplify inside the curvy parentheses, then I'll simplify inside the square brackets, and only then will I take care of the squaring. After that is done, then I can finally add in the 4 :

4 + [−1(−2 − 1)] 2

= 4 + [−1(−3)] 2

= 4 + [3] 2

How do I know what grouping symbols to use?

There is no particular significance in the use of square brackets (the "[" and "]" above) instead of parentheses. Brackets and curly-braces (the "{" and "}" characters) are often used when there are nested parentheses, as an aid to keeping track of which close-parentheses go with which open-parens. The different grouping characters are used for convenience only.

This is similar to what happens in a spreadsheet when you enter a formula using parentheses: each set of parentheses is color-coded, so you can tell the pairs:

I will simplify inside the parentheses first, and only then multiply by the 4 . Yes, this means that I'll be adding before doing the multiplication, but that's okay; the addition is inside grouping symbols, so it comes first.

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Rules and properties

There are many mathematical rules and properties that are necessary or helpful to know when trying to solve math problems. Learning and understanding these rules helps students form a foundation they can use to solve problems and tackle more advanced mathematical concepts.

Basic mathematical properties

Some of the most basic but important properties of math include order of operations, the commutative, associative, and distributive properties, the identity properties of multiplication and addition, and many more. They are properties that are used throughout most areas of mathematics in some form or other.

Order of operations

Order of operations is often taught using one of two acronyms: PEMDAS or BODMAS. Both indicate the order in which operations should be carried out.

PEMDAS: Parentheses, exponents, multiplication, division, addition, subtraction

BODMAS: Brackets, order, division, multiplication, addition, subtraction

Notice that multiplication and division are in different positions in PE(MD)AS and BO(DM)AS; this is because multiplication and division, and addition and subtraction, can be performed in either order, and usually when deciding the order of performing these operations (assuming parentheses and exponents are already taken into account), they are carried through from left to right.

The mnemonic device "Please excuse my dear aunt Sally" is commonly used as a way to remember the acronym PEMDAS. It is not necessary if you can remember the acronym immediately, but can be helpful to remember just in case.

Commutative property

The commutative property states that changing the order in which two numbers are added or multiplied does not change the result.

a + b = b + a

a × b = b × a

Associative property

The associative property states that changing the way that numbers are grouped in addition and multiplication does not change the result.

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Distributive property

The distributive property states that multiplying a group of numbers that are being added is the same as multiplying by each number individually, then adding them together.

a × (b + c) = a × b + a × c

Order of Operations

The order of operations is a set of rules that is to be followed in a particular sequence while solving an expression. In mathematics with the word operations we mean, the process of evaluating any mathematical expression, involving arithmetic operations such as division, multiplication, addition, and subtraction. Let us learn in detail about the order of operations rules and how well we can remember the rules using short tricks.

What is the Order of Operations?

The order of Operations is the rule in math that states we evaluate the parentheses/brackets first, the exponents/the orders second, division or multiplication third (from left to right, whichever comes first), and the addition or subtraction at the last (from left to right, whichever comes first). In math, there might be several operations to be done while evaluating an expression, and simplification at the end yields different results. However, we can only have one correct answer for any sort of expression. To identify the correct answer we simplify any given mathematical expression using a certain set of rules. These rules revolve around all the basic operators used in maths. Operators such as addition (+), subtraction (-), division (÷), and multiplication (×). Look at the given image to get a glimpse of how the order of operations exactly looks like.

Order of Operations Definition

As we discussed above Order of operations can be defined as, a set of basic rules of precedence we use while solving any mathematical expression, involving multiple operations. When a subexpression appears between two operators, the operator that comes first according to the list given below should be applied first. The order of operations, rules are expressed here:

• Brackets ( ), { }, [ ]
• Division (÷) and Multiplication (×)
• Addition (+) and Subtraction (-)

The above-mentioned set of rules always varies according to the respective given mathematical expressions.

Order of Operations Rules

While performing any sort of an operation on the respective numbers present in the expression we will follow the given basic rules in the particular sequence.

Order of Operations Rule 1: Observe the expression. The first rule is to solve the numbers present inside the parentheses or brackets. We solve inside to out grouping operations. Note the pattern of brackets present in the expression, there is a particular order to solve the parentheses, i.e., [ { ( ) } ]. First, solve the round brackets ( ) → curly brackets { } → box brackets [ ]. Inside the parantheses the order of operations are to be followed. Order of Operations Rule 2: After solving the numbers in the parentheses, look for any term present in the form of exponents and solve it. Order of Operations Rule 3: Now we are left with the basic four operators. Look for the numbers with the operation of multiplication or division, solve them from left to right. Order of Operations Rule 4: Lastly, look for the terms with addition or subtraction and solve them from left to right.

These rules have a specific acronym name. We call them PEMDAS or BODMAS . Let us learn now what exactly PEMDAS or BODMAS is.

Order of Operations - PEMDAS vs BODMAS

The PEMDAS or BODMAS is the two different acronym names given to learn the rules. These two names state the order in which the operations in an expression should be followed. Here is the detailed term for each letter used in the mentioned acronyms. First, we will discuss the PEMDAS.

Order of Operations PEMDAS

• P stands for Parentheses ( ), { }, [ ]
• E stands for Exponents (a 2 ) (For example, here, a is a number with exponent 2 )
• M stands for Multiplication (×)
• D stands for Division (÷)
• A stands for Addition (+)
• S stands for Subtraction (-)

Order of Operations BODMAS

• B stands for Brackets ( ), { }, [ ]
• O stands for Order

With the help of the above denotations, we can easily solve the mathematical expressions and get the correct answer.

How to Use Order of Operations?

Let us look at the different examples mentioned below to understand the accuracy of the rules used in order of operations.

1) For solving parentheses in order of operations :

Expression: 4 × (5 + 2) Solution: 4 × ( 7 ) = 28 (Correct (✔). This is a correct way to solve the parentheses) Let us look at another approach for the same expression. 4 × ( 5 + 2) = 20 + 2 = 22 (Incorrect (✘). This is an incorrect way to solve the parentheses)

2) For solving exponents in order of operations

Expression: 4 × (5 2 ) Solution: 4 × ( 25 ) = 100 (Correct (✔). This is a correct way to solve the exponents) Let us look at another approach for the same expression. 4 × ( 5 2 ) = 20 2 = 400 ((Incorrect (✘). This is an incorrect way to solve the exponents)

3) For multiplication or division and addition or subtraction

Expression: 3 + 5 × 2 Solution: 3 + 5 × 2 = 3 + 10 = 13 (Correct (✔). This is a correct way.) Let us look at another approach for the same expression. 3 + 5 × 2 = 8 × 2 = 16 (Incorrect (✘). This is an incorrect way.)

Expression: 3 - 6 ÷ 2 Solution: 3 - 6 ÷ 2 = 3 - 3 = 0 (Correct (✔). This is a correct way.) Let us look at another approach for the same expression. 3 - 6 ÷ 2 = (-3) ÷ 2 = -3/2 (Incorrect (✘). This is an incorrect way.)

Always remember while following the rules of order of operations do multiplication or division before addition or subtraction

Ways to Remember Order of Operations

We just read about the two different words PEMDAS and BODMAS. This is the best way to remember the order of operations. PEMDAS can be remembered by the phrase "Please Excuse My Dear Aunt Sally". In the order of operations, it means "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". Here multiplication and division, addition and subtraction are together. Similarly, we can remember the order of operations with the word BODMAS (Brackets, Orders, Division, Multiplication, Addition, and Subtraction.).

The easiest way to learn the order of operation is to perform the given steps:

• Start simplifying terms within the brackets first
• Solve the exponential terms.
• Perform division or multiplication.
• Perform addition or subtraction.

Note: While performing the order of operations on any given expression, we must observe the pattern of operators.

Real-Life Applications of Order of Operations

A lot of activities in our life require some sort of order of operation to perform it well. Let us take an everyday problem. Suppose you went to purchase five pepperoni pizzas that cost $20 each, and you want to split the total cost among 5 people evenly. To find out how much each person needs to pay let's use the order of operations here.

Total number of people = 5 Total number of pizzas = 5 Cost of one pizza = $20 Let us frame an expression using PEMDAS: Expression: (20 + 20 + 20 + 20 + 20) ÷ 5 or (5 × 20) ÷ 5 Solution: According to PEMDAS or BODMAS we will first solve the parentheses. (100) ÷ 5 = 20 According to the order of operations, each person needs to pay $20.

Similar to the above-mentioned problem, we have many day-to-day real-life instances where we use order of operations to deal with our problems.

☛Related Articles on Order of Operations

Check out the interesting articles below and learn more about the topic Order of Operations and its applications in detail.

• Subtraction
• Multiplication
• Order of Operations Worksheets 5th Grade

Order of Operations Examples

Example 1: Help Jack in solving the following problem with the help of order of operations rules. a) 18 ÷ (9 - 2 × 3)

Solution : Given expression: 18 ÷ (9 - 2 × 3) According to the order of operations rule, we have to solve parentheses first. Please note here inside the parentheses we have two operations present, multiplication, and subtraction. First, multiply 2 × 3 = 6 18 ÷ (9 - 6) Now subtract 6 from 9, 18 ÷ (3) Now divide 18 ÷ 3 = 6

Example 2: Simplify the given expression using the order of operations rules. ​(6 × 2 - 6 - 1) × 2 2

Solution: We know that the order of operation follows either PEMDAS or BODMAS . Let us follow the order of operations rules and simplify the given expression.

Step 1: First, we need to solve the numbers within the parentheses. Multiply 6 by 2 in the given expression, ​​​​​(6 × 2 - 6 - 1) × 2 2 , we get, (12 - 6 - 1) × 2 2 . Step 2- Now, we need to subtract 6 from 12 inside the bracket, so, we get, (6 - 1) × 2 2 . Step 3- Remove parentheses after subtracting 6 - 1, we get, 5 × 2 2 . Step 4- Solve exponent, i.e 2 2 = 4. Step 5- Multiply 5 by 4 to get the final answer, which is, 5 × 4 = 20. ∴ ​(6 × 2 - 6 - 1) × 2 2 = 20 .

Example 3: Evaluate the expression using the order of operations: (1 + 20 − 9 ÷ 3 2 ) ÷ ((2 + 1) 2 + 16 ÷ 2)

Solution: Let us see how we can apply the rules of the order of operations in solving the given expression. Step 1: First, we need to simplify the innermost bracket, (1 + 20 − 9 ÷ 3 2 ) ÷ (3 2 + 16 ÷ 2) Step 2: Now we have to evaluate exponents, (1 + 20 − 9 ÷ 9) ÷ (9 + 16 ÷ 2) Step 3: Now, we need to divide 9 by 9 and 16 by 2 inside the brackets, and we get, (1 + 20 − 1) ÷ (9 + 8) Step 4: Adding 1 and 20 we get 21. Now subtract 1 from 21 we get 20. Now, (20) ÷ (9 + 8) Step 5: Add 9 + 8 and divide the result by 20. 20 ÷ 17 = 20/17 Step 6: ∴ (1 + 20 − 9 ÷ 3 2 ) ÷ ((2 + 1) 2 + 16 ÷ 2) = 20/17

Example 4: Solve the statement problem using the order of operations. If 72 is divided by the sum of 4 and 5, then subtracted from 10, what will be the final answer?

Solution: Let us first write the given statement into mathematical form. 10 - [72 ÷ (4 + 5)] Using the order of operations rules this expression can be simplified as: = 10 - [72 ÷ (4 + 5)] = 10 - [72 ÷ 9] = 10 - 8 = 2 ∴ 10 - [72 ÷ (4 + 5)] = 2

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Practice Questions on Order of Operations

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FAQs on Order of Operations

What is the order of operations in math.

The order of operation in math is a set of rules revolving around 4 major operators. According to the order of operations, there is a particular sequence which we need to follow on each operator while solving the given mathematical expression.

How to Solve Order of Operations?

To solve the order of operations first observe the expression and note what pattern it exactly follows. Now start using either PEMDAS or BODMAS to solve the given expression. As per the rules of order of operations do look for parentheses first, then exponents, then move towards multiplication or division and addition or subtraction from left to right.

How to Do Order of Operations with Integers?

We know that integers are positive numbers and negative numbers. We can easily perform the order of operations with integers by following the given steps:

• Look for the integers present inside the parentheses or brackets and solve them.
• After solving the integers in the parentheses, look for any integer term present in the form of exponents and solve it.
• Now we are left with the basic four operators to be performed on integers. Look for the integers with the operation of multiplication or division and solve them from the left-hand side to the right-hand side.
• Lastly, look for the integers with addition or subtraction and solve them.
• In the case of integers, we need to make sure we are properly multiplying the signs. Such that, (-) × (-) = + and (+) × (-) = -

How to Remember Order of Operations?

To remember the order of operations we use two famous acronyms, i.e. PEMDAS and BODMAS. We use either of the two according to the rules of order of operations. PEMDAS or BODMAS helps in remembering the process of solving any order of operation for any n number of expressions.

How to do Order of Operations with Exponents?

According to the PEMDAS, the letter E stands for exponents which come as the second step in order of operations. Let us look at the given example to clearly understand how to do the order of operations with exponents. Expression: 7 × (2 2 ) Solution: 7 × ( 4 ) = 28 (Correct (✔). This is a correct way to solve the exponents) Let us look at another approach for the same expression. 7 × ( 2 2 ) = 14 2 = 196 ((Incorrect (✘). This is an incorrect way to do order of operations with exponents)

What is the Correct Order of Operations?

The correct order of operations can be easily expressed by using the word PEMDAS or BODMAS. The two words can be described as PEMDAS (Parentheses, Exponents, Multiplication or Division, and Addition or Subtraction). Similarly, for BODMAS (Brackets, Orders, Division, Multiplication, Addition, and Subtraction.)

What is the Order of Operations without Parentheses?

Going by the rules of the order of operations if we remove the parentheses, then we are left with EMDAS. EMDAS stands for (Exponents, Multiplication or Division, and Addition or Subtraction). If in the expression we don't have any exponential term then we need to perform multiplication or division first and moving forward we proceed with addition or subtraction. The situation may vary according to the operators present in the given expression

When Do We Use Order of Operations?

A lot of instances in our life pass through some sort of order of operations to perform it well. Every day we encounter such a scenario. For example, going to the grocery market and purchasing things we quickly perform the order of operations in our head. This helps us in reducing the turnaround time at the billing counter.

What Operation Is Completed First In the Order of Operations?

In the above sections, we read about two acronyms BODMAS and PEMDAS. According to both the acronyms, in order of operations, we simplify parentheses or the brackets first.

What is the Use of Order of Operation Calculator?

Order of operations calculator is an online tool and the fastest method with which we can evaluate any given numerical expression keeping the order of operations rules in mind. To use the order of operation calculator we need to enter the numerical expression in the correct format. Try Cuemath's order of operations calculator and solve the expressions quickly within a seconds.

☛Also Check:

For more practice try these:

• Order of Operations With Exponents Worksheets
• Advanced Order of Operations Worksheets
• Order of Operations Worksheets

What Are the 4 Order of Operations?

The 4 major order of operations are:

• Parentheses.
• An exponential term.
• Multiplication or division .
• At the end addition or subtraction.

The four order of operations can be easily recalled at any given point in time by learning the acronyms PEMD AS or B ODMAS .

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• Order of operations introduction

Worked example: Order of operations (PEMDAS)

• Order of operations (no exponents)

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Video transcript

Order Of Operations – Definition, Steps, FAQs, Examples

What is the order of operations in math.

• Order of Operations Steps

Solved Examples On Order Of Operations

Practice problems on order of operations, frequently asked questions on order of operations.

There are many operations in mathematics, such as addition , subtraction , multiplication , and division . They help us evaluate mathematical expressions.

Consider the following expression: 4+ 5 × 32 – 2

The expression consists of many operations. But which part do you calculate first?

You may start from the left and get one answer. But your friend may begin from the right and get a completely different answer!

Note: Both the methods given above are incorrect.

Hence, to avoid confusion, a standard rule was set to perform such calculations. This rule is known as the order of operations.

If you have an expression where all the operations are the same (example: only addition, only subtraction, only multiplication, or only division) then the correct way to solve it would be from left to right. But for expressions with multiple operations, we need to follow the order of operations.

The order of operations is the rule that tells us the sequence in which we should solve an expression with multiple operations.

A way to remember that order is PEMDAS. Each letter in PEMDAS stands for a mathematical operation. 

Order of Operations Steps:

Parentheses.

The first step is to solve the operation within parentheses or brackets . Parentheses are used to group things together. Work out all groupings from inside to out.

Work out the exponential expressions after the parentheses.

Multiplication and Division

Next, moving from left to right, multiply and/or divide, whichever comes first.

Addition and Subtraction

Lastly, moving from left to right, add and/or subtract, whichever comes first.

Related Worksheets

Why Follow the Order of Operations?

We follow the rules of the order of operations to solve expressions so that everyone arrives at the same answer. 

Here’s an example of how we can get different answers if the correct order of operations is NOT followed:

Example 1: Solve: 2 + 6 × (4 + 5) ÷ 3 – 5 using PEMDAS.

Step 1 – Parentheses : 2+6 × (4 + 5) ÷ 3 – 5 = 2 + 6 × 9 ÷ 3 – 5

Step 2 – Multiplication: 2 + 6 × 9 ÷ 3 – 5 = 2 + 54 ÷ 3 – 5

Step 3 – Division: 2 + 54 ÷ 3 – 5 = 2 + 18 – 5

Step 4 – Addition: 2 + 18 – 5 = 20 – 5

Step 5 – Subtraction: 20 – 5 = 15

Example 2: Solve 4 – 5 ÷ (8 – 3) × 2 + 5 using PEMDAS.

Step 1 – Parentheses: 4 – 5 ÷ (8 – 3) × 2 + 5 = 4 – 5 ÷ 5 × 2 + 5

Step 2 – Division: 4 – 5 ÷ 5 × 2 + 5 = 4 – 1 × 2 + 5

Step 3 – Multiplication:  4 – 1 × 2 + 5 = 4 – 2 + 5

Step 4 – Subtraction: 4 – 2 + 5 = 2 + 5

Step 5 – Addition: 2 + 5 = 7

Example 3: Solve 100 ÷ (6 + 7 × 2) – 5 using PEMDAS.

Step 1 – Multiplication inside parentheses: 100 ÷ (6 + 7 × 2 ) – 5= 100 ÷ (6 + 14) – 5

Step 2 – Addition inside parentheses: 100 ÷ (6 + 14) – 5 = 100 ÷ 20 – 5

Step 3 – Division: 100 ÷ 20 – 5 = 5 – 5 Step 4 – Subtraction: 5 – 5 = 0

Order Of Operations - Definition With Examples

Attend this Quiz & Test your knowledge.

Simplify 4+ (5 ×3² + 2) using PEMDAS.

Simplify 9 – 24 ÷ 8 × 2 + 3 using pemdas., simplify [(32 ÷ 4) + 3] × 2 using pemdas., simplify $(3 × 5² ÷ 5)$ – $(16 - 10)$ using pemdas..

What is the order of operations in math?

The order of operations are the rules that tell us the sequence in which we should solve an expression with multiple operations.

The order is PEMDAS: Parentheses, Exponents, Multiplication, and Division (from left to right), Addition and Subtraction (from left to right).

Is there a trick we can use to remember the order of operations?

Yes. You can use the phrase “Please Excuse My Dear Aunt Sally” to remember PEMDAS.

Can we perform subtraction before addition?

Yes, addition and subtraction are at the same level according to the PEMDAS rule. So, without brackets, we do the math from left to right if we are only dealing with addition and subtraction. For example, 9 – 6 + 3 = 3 + 3 = 6.

Can we perform division before multiplication?

Yes, multiplication and division are at the same level according to the PEMDAS rule. So, without brackets, we do the math from left to right if we are only dealing with multiplication and division. For example, 24 ÷ 8 × 2 = 3 × 2 = 6.

Are PEMDAS and BODMAS the same?

Yes. Both PEMDAS and BODMAS are acronyms for remembering the order of operations. They are different names for the same rule. What they call PEMDAS in the US is called BODMAS in the UK, Australia, India and various other countries.

NOTE – Related Readings:

• Multiplication

RELATED POSTS

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• Subtract Fractions with Unlike Denominators – Definition with Examples
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Order of Operations Worksheets - Addition, Subtraction, Multiplication & Division

Related Topics & Worksheets: Order Of Operations Order Of Operations Worksheet

Objective: I know how to perform mixed operations with addition, subtraction, multiplication and division. If the calculations involve a combination of addition, subtraction, multiplication and division then

Step 1: First, perform the multiplication and division from left to right.

Step 2: Then, perform addition and subtraction from left to right. Example: Calculate 9 × 2 – 10 ÷ 5 + 1 =

Solution: 9 × 2 – 10 ÷ 5 + 1 (perform multiplication)

= 18 – 10 ÷ 5 + 1 (perform division)

= 18 – 2 + 1 (perform subtraction)

= 16 + 1 (perform addition)

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Sequences - Finding a Rule

To find a missing number in a Sequence, first we must have a Rule

A Sequence is a set of things (usually numbers) that are in order.

Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for a more in-depth discussion.

Finding Missing Numbers

To find a missing number, first find a Rule behind the Sequence.

Sometimes we can just look at the numbers and see a pattern:

Example: 1, 4, 9, 16, ?

Answer: they are Squares (1 2 =1, 2 2 =4, 3 2 =9, 4 2 =16, ...)

Rule: x n = n 2

Sequence: 1, 4, 9, 16, 25, 36, 49, ...

Did you see how we wrote that rule using "x" and "n" ?

x n means "term number n", so term 3 is written x 3

And we can calculate term 3 using:

x 3 = 3 2 = 9

We can use a Rule to find any term. For example, the 25th term can be found by "plugging in" 25 wherever n is.

x 25 = 25 2 = 625

How about another example:

Example: 3, 5, 8, 13, 21, ?

After 3 and 5 all the rest are the sum of the two numbers before ,

That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence :

3, 5, 8, 13, 21, 34, 55, 89, ...

Which has this Rule:

Rule: x n = x n-1 + x n-2

Now what does x n-1 mean? It means "the previous term" as term number n-1 is 1 less than term number n .

And x n-2 means the term before that one .

Let's try that Rule for the 6th term:

x 6 = x 6-1 + x 6-2

x 6 = x 5 + x 4

So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:

x 6 = 21 + 13 = 34

One of the troubles with finding "the next number" in a sequence is that mathematics is so powerful we can find more than one Rule that works.

What is the next number in the sequence 1, 2, 4, 7, ?

Here are three solutions (there can be more!):

Solution 1: Add 1, then add 2, 3, 4, ...

So, 1+ 1 =2, 2+ 2 =4, 4+ 3 =7, 7+ 4 =11, etc...

Rule: x n = n(n-1)/2 + 1

Sequence: 1, 2, 4, 7, 11, 16, 22, ...

(That rule looks a bit complicated, but it works)

Solution 2: After 1 and 2, add the two previous numbers, plus 1:

Rule: x n = x n-1 + x n-2 + 1

Sequence: 1, 2, 4, 7, 12, 20, 33, ...

Solution 3: After 1, 2 and 4, add the three previous numbers

Rule: x n = x n-1 + x n-2 + x n-3

Sequence: 1, 2, 4, 7, 13, 24, 44, ...

So, we have three perfectly reasonable solutions, and they create totally different sequences.

Which is right? They are all right.

Simplest Rule

When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.

Finding Differences

Sometimes it helps to find the differences between each pair of numbers ... this can often reveal an underlying pattern.

Here is a simple case:

The differences are always 2, so we can guess that "2n" is part of the answer.

Let us try 2n :

The last row shows that we are always wrong by 5, so just add 5 and we are done:

Rule: x n = 2n + 5

OK, we could have worked out "2n+5" by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.

Second Differences

In the sequence {1, 2, 4, 7, 11, 16, 22, ...} we need to find the differences ...

... and then find the differences of those (called second differences ), like this:

The second differences in this case are 1.

With second differences we multiply by n 2 2

In our case the difference is 1, so let us try just n 2 2 :

We are close, but seem to be drifting by 0.5, so let us try: n 2 2 − n 2

Wrong by 1 now, so let us add 1:

The formula n 2 2 − n 2 + 1 can be simplified to n(n-1)/2 + 1

So by "trial-and-error" we discovered a rule that works:

Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...

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