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The PEMDAS Rule: Understanding Order of Operations

General Education

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.

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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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Order of Operations PEMDAS

"Operations" mean things like add, subtract, multiply, divide, squaring, etc. If it isn't a number it is probably an operation.

But, when you see something like ...

7 + (6 × 5 2 + 3)

... what part should you calculate first? Start at the left and go to the right? Or go from right to left?

Warning: Calculate them in the wrong order, and you can get a wrong answer !

So, long ago people agreed to follow rules when doing calculations, and they are:

Order of Operations

Do things in Parentheses First

Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract

Multiply or Divide before you Add or Subtract

Otherwise just go left to right

How Do I Remember It All ... ? PEMDAS !

Divide and Multiply rank equally (and go left to right).

Add and Subtract rank equally (and go left to right)

So do it this way:

After you have done "P" and "E", just go from left to right doing any "M" or "D" as you find them.

Then go from left to right doing any "A" or "S" as you find them.

You can remember by saying " P lease E xcuse M y D ear A unt S ally".

You may prefer GEMS ( G rouping, E xponents, M ultiply or Divide, Add or S ubtract). In the UK they say BODMAS (Brackets, Orders, Divide, Multiply, Add, Subtract). In Canada they say BEDMAS (Brackets, Exponents, Divide, Multiply, Add, Subtract). It all means the same thing! It doesn't matter how you remember it, just so long as you get it right.

Example: How do you work out 3 + 6 × 2 ?

M ultiplication before A ddition:

First 6 × 2 = 12 , then 3 + 12 = 15

Example: How do you work out (3 + 6) × 2 ?

P arentheses first:

First (3 + 6) = 9 , then 9 × 2 = 18

Example: How do you work out 12 / 6 × 3 / 2 ?

M ultiplication and D ivision rank equally, so just go left to right:

First 12 / 6 = 2 , then 2 × 3 = 6 , then 6 / 2 = 3

A practical example:

Example: Sam threw a ball straight up at 20 meters per second, how far did it go in 2 seconds?

Sam uses this special formula that includes the effects of gravity:

height = velocity × time − (1/2) × 9.8 × time 2

Sam puts in the velocity of 20 meters per second and time of 2 seconds:

height = 20 × 2 − (1/2) × 9.8 × 2 2

Now for the calculations!

The ball reaches 20.4 meters after 2 seconds

Exponents of Exponents ...

What about this example?

Exponents are special: they go top-down (do the exponent at the top first). So we calculate this way:

So 4 3 2 = 4 (3 2 ) , not (4 3 ) 2

And finally, what about the example from the beginning?

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. 

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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.

Recommended Worksheets

More 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

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What is the Order of Operations in Math (PEMDAS)?

Does the order in which we solve math problems matter? Yes! Let’s learn about the order of operations in math, aka PEMDAS, to solve multi-step expressions.

Author Amber Watkins

Published November 7, 2023

Published Nov 7, 2023

• Key takeaways
• The order of operations teaches the order in which to solve multi-operational expressions.
• The order of operations can easily be remembered using the acronym PEMDAS .
• As your child successfully masters the order of operations , they will be able to solve more advanced problems with ease.

Table of contents

• What is the order of operations?
• Order of operations rules

When reading a math problem, we usually start from left to right. So it is natural to want to solve problems in the same way. Does this method get us the right answer for every problem?

Let’s solve the following problem in two ways to see how order makes a difference: 3 – 2 x 5 =

1. You can solve this problem by subtracting first.

3 – 2 x 5

2. Or you can solve it by multiplying first.

3 – 10

In both examples, we took the same actions: we multiplied and subtracted , but switched the order. As a result, we found two very different answers. How can we know which is the right answer? By following the math order of operations .

What is the Order of Operations in Math?

Operations are the actions we take in math. The most common math operations are adding , subtracting, multiplying, dividing , simplifying exponents, and finding square roots .

Some math problems have more than one kind of operation in the problem. How can we solve those types of problems? We follow the order of operations ! What is the order of operations? The order of operations tells us the order in which to perform operations when the problem contains more than one kind of operation.

Order of Operations Rules

Many times you will see the order of operations called PEMDAS. What is PEMDAS? PEMDAS is an acronym that helps us remember the order in which to solve multi-operational problems. What does PEMDAS stand for? PEMDAS is an acronym that stands for Parenthesis, Exponents, Multiplication or Division, Adding or Subtracting.

Now that we know the Pemdas meaning , let’s discuss the Pemdas Rule . The most important PEMDAS Rule is when solving order of operation problems in math, you must follow the order of the acronym.

• First, you solve whatever operations are in the parenthesis . ()
• Second, you calculate any exponents . x²
• Third, you multiply or divide . x  or  ÷
• Fourth, you add or subtract . + or – 

BODMAS vs PEMDAS

You may also hear the order of operations called BODMAS. What is BODMAS?  BODMAS stands for Brackets, Orders (Powers or Roots), Division or Multiplication, Adding or Subtracting.

Brackets are another name for parenthesis and orders are another name for exponents. Although the terms used for operations are different than Pemdas, both Bodmas and Pemdas follow the same rules of order of operations . What are those rules and how can we use them to solve PEMDAS problems ?

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How to Solve Order of Operations Problems

Let’s solve this Order of Operations problem as we learn some PEMDAS rules. 4² – ( 3 x 5 ) + 9 ፥ 3 x 2 

In our example problem, we would multiply 3 x 5 first because it is in the parenthesis . P E M D A S 

4² – ( 3 x 5 ) + 9 ፥ 3 x 2

4² – 15 + 9 ፥ 3 x 2 2. Pemdas Rule for Exponents Exponents should always be simplified before moving to the next step in the order of operations. If you begin multiplying, dividing, adding, or subtracting while there is still an exponent in the problem, you may have missed this step. This is how we would simplify the exponents . P E M D A S 

4² – 15 + 9 ፥ 3 x 2 16 – 15 + 9 ፥ 3 x 2 3. Pemdas Rule for Multiplying or Dividing When using the order of operations, you may have noticed the word “or” is used when multiplying or dividing. When following the order of operations in math , you can multiply or divide in any order. Simply complete the operation that comes first. We have both multiplication and division in this problem, so we will go from left to right. First is division, then is multiplication. 

P E M D A S 

16 – 15 + 9 ፥ 3 x 2

16 – 15 + 3 x 2

16 – 15 + 6

4. Pemdas Rule for Adding or Subtraction

When using the order of operations, you may have also noticed the word “or” is used when adding or subtracting. When following the order of operations in math , you can add or subtract in any order. Simply complete the operation that comes first.

Finally, we would add or subtract depending on which operation comes first.

16 – 15 + 6 

Now that we know some of the PEMDAS rules, let’s go over some PEMDAS examples together. 

PEDMAS Examples

PEMDAS EXAMPLE 1: Solve 3 + ( 8 x 9 ) =

Explanation : First, we see parenthesis comes first in the P EMDAS acronym. So we would multiply 8 x 9 because it is in the parenthesis. Therefore, 8 x 9 is equal to 72. Then we would add 3 and 72 to get 75 as our answer.

Our Work for this PEMDAS example would look like this:         

3 + ( 8 x 9 ) 3 + 72 75 

PEMDAS EXAMPLE 2: Solve 12  ፥ 3 + 14 =

First, we see that division comes before adding in PEM D AS. So we divide 12 by 3 which is equal to 4. Next, add 4 and 14 which equals 18. Our Work for this PEMDAS example would look like this:   12 ፥ 3 + 14 4 + 14 18

PEDMAS Practice Questions

Scroll down for the answers!

1. Use Pemdas to solve 2² + 12 – ( 3 x 2) =

2. Use Pemdas to Solve (3 x 2) + (12 ፥ 3 ) =

3. Use Pemdas to Solve 100  ፥  10 – (6 ፥ 2 ) =

FAQs About Order of Operations

Pemdas stands for Parenthesis, Exponents, Multiplication or Division, Adding or Subtracting.

In PEMDAS problems , multiplication or division can happen in any order. The Pemdas rule is to complete whichever operation comes first.

In Pemdas problems , always complete whichever operation is inside of the parenthesis first.

BODMAS stands for Brackets, Orders, Division or Multiplication, Adding or Subtracting.

Yes. PEMDAS and BODMAS are both acronyms to represent the order of operations, but they use different words to describe the same operation. For example, PEMDAS uses the terms parenthesis and exponents. BODMAS uses the terms brackets and powers. 

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Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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First, multiply 3 x 2 because it is in the parenthesis. Then, simplify the exponent 2² which equals 4. Next, add 4 + 12 which equals 16. Finally, subtract 6 which equals 10. 

2² + 12 – (3 x 2) =                                                            2² + 12 – 6 = 4  + 12 – 6 = 16 – 6 = 10

First, multiply 3 x 2 because it is in the first parenthesis. Next, divide 12 by 3 because it is in the second parenthesis. Finally, add 6 + 4 which equals 10. (3 x 2) + (12 ፥ 3 ) 6 + (12 ፥ 3 ) 6 + 4   10

First, divide 6 by 2 because it is in the parenthesis. Next, divide 100 by 10. Finally, subtract 3 from 10 which equals 7. 

100  ፥  10 – (6 ፥ 2 )    100  ፥  10 – 3 10 – 3 7

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Order of Operations (PEMDAS)

Order of operations.

The fundamental concept behind the order of operations is to perform arithmetic operators in the “right” order or sequence. Let’s take a look at how Rob and Patty tried to simplify a given numerical expression by applying the order or rule of operations.

What is Rob’s mistake?

• He carelessly simplified the numerical expressions by applying arithmetic operations from left to right.

Patty got the correct answer because she properly applied the rules on order of operations.

• She performed multiplication and division first before addition and subtraction.

What is the Order of Operations?

The order of operations is merely a set of rules that prioritize the sequence of operations starting from the most important to the least important.

This rule on how to correctly simplify numerical expressions is also known as the PEMDAS Rule (an acronym for P lease E xcuse M y D ear A unt S ally).

Step 1: Do as much as you can to simplify everything inside the parenthesis or grouping symbol.

Step 2: Simplify exponential numbers in the numerical expression, wherever possible.

Step 3: Multiply and divide whichever comes first, from left to right

Step 4: Add and subtract whichever comes first, from left to right

Examples on How to Apply the Order of Operations to Simplify Numerical Expressions

Example 1: Simplify the expression below using the Order of Operations.

• Examining the numerical expressions with multiple operations from left to right, we can see that we must perform division first, which is [latex]5 \div 5 = 1[/latex].

• At this point, I have three (3) possible operations. In the Order of Operations, multiplication takes precedence over addition and subtraction. Therefore, we should multiply next. We have [latex]6 \times 2 = 12[/latex].

• What should we do next, add or subtract? Based on the Order of Operations, addition and subtraction are of equal importance. To determine which operation to perform first, we add or subtract whichever comes first from left to right which in this situation is to add, [latex]1 + 3 = 4[/latex].

• Now we are left with one operation, subtraction. The seemingly complex numerical expression is reduced to the final answer of [latex] – 8[/latex].

Example 2: Simplify the expression below using the Order of Operations.

The next examples will involve parentheses . Remember that you have to simplify everything inside the parenthesis first before going forward.

Example 3: Simplify the expression below using the Order of Operations.

• Take a look at the expressions inside the parenthesis. The rule tells us to divide first before we subtract.

• We can get rid of the parenthesis by subtracting [latex]7[/latex] by [latex]2[/latex].

• Multiplication is much “stronger” operation that subtraction that’s why we must first multiply [latex]5[/latex] and [latex]4[/latex].

• Finish it off by subtracting [latex]25[/latex] by [latex]20[/latex].

Example 4:   Simplify the expression below using the Order of Operations.

• Simplify the expressions inside the parentheses first. Multiply on the first parenthesis and divide on the second.

• Add the numbers inside the first parenthesis then subtract the numbers inside the second.

• We have multiplication and division here. Since multiplications comes before division, we are going to multiply first.

• Between subtracting and dividing, division takes precedence therefore we divide [latex]5[/latex] by [latex]5[/latex] to get [latex]1[/latex].

• The last operation left is subtraction therefore that’s what we’re going to do.

The final examples will involve exponents so be careful with each step because they are so many things going on. As long as you remain focus in following the rules governing the order of operations, it shouldn’t be that difficult! Here we go…

Example 5: Simplify numerical expression below using the rules of Order of Operations.

• Simplify the expressions inside the parentheses. But more specifically, simplify the numbers with exponents.

• There are two parentheses to simplify. We will simplify the second one which is [latex]\left( {30 – 27} \right)[/latex] because it is much simpler. Here, the difference of [latex]30[/latex] and [latex]27[/latex] is [latex]3[/latex].

• Now we turn our attention to the other parenthesis. The Order of Operations tells us to divide before we subtract.

• Finally we can get rid of the parenthesis by performing subtraction as there is no other thing to do.

• Looking at what we have left, simplifying exponential numbers takes precedence over the operations of multiplication, addition, and subtraction.

• Scanning from left to right, it’s obvious that we must multiply before adding and/or subtracting.

• Based on the Order of Operations, addition and subtraction are of equal importance. We must subtract first because the operation of subtraction comes before addition when viewed from left to right.

• The last step is to add because there is no other operations left.

Example 6: Simplify the expression below using the Order of Operations.

• Focus our attention on the single grouping symbol which is the parenthesis. Notice that there is a number with an exponent. Let’s simplify [latex]{3^3}[/latex] first.

• Looking inside the parenthesis, we must divide first before multiplying and subtracting.

• Keeping still our attention inside the parenthesis, the order of operations tells us to multiply before we subtract.

• The last operation left inside the parenthesis is subtraction. Let’s perform that!

• Let’s take a pause here. At this point, it is clear that we can perform three (3) simplifications simultaneously. It’s obvious to see that we have three exponential expressions which are [latex]{2^5}[/latex], [latex]{4^2}[/latex], and [latex]{3^2}[/latex].

• The seemingly complex problem is now reduced to something that is very easy to simplify. Looking at it from left to right, division takes precedence over subtraction and addition.

• Because subtraction and multiplication are on the same level in the hierarchy of operations. The way we break the tie as you should already know it is to perform the one that comes first when viewed from left to right. In this situation, we will subtract then add. That’s it!

Take a quiz:

• Order of Operations Quiz

You might also like these tutorials:

• Order of Operations Practice Problems with Answers
• PEMDAS Rule

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PEMDAS: Remembering Math's Order of Operations

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Nearly every middle school in the U.S. teaches its students to remember this simple phrase: "Please excuse my dear Aunt Sally." But why are we apologizing for her behavior? Did she wear white after Labor Day or something?

The world may never know. In all seriousness, " P lease E xcuse M y D ear A unt S ally," or PEMDAS , is just a mnemonic . It's a tool educators use to help us memorize information through a catchy rhyme, phrase or acronym. Now let's explore how to use this tool to solve equations.

What Is PEMDAS?

Origins of the order of operations, why do we use pemdas, solving math equations using the pemdas rule, beyond pemdas.

PEMDAS is an acronym and mnemonic that represents a set of rules used to clarify the order in which operations should be performed to correctly evaluate mathematical expressions. PEMDAS stands for:

• Parentheses : This means any calculations inside parentheses should be performed first. This can include brackets or other grouping symbols.
• Exponents : This refers to powers or square roots. You handle these calculations after dealing with parentheses but before other operations.
• Multiplication : When you encounter multiplication in an expression, after parentheses and exponents, you perform this operation next.
• Division : Similar to multiplication, you handle division after parentheses and exponents, working from left to right.
• Addition : After the aforementioned operations, you perform addition.
• Subtraction : Lastly, after all the other operations have been handled, you perform subtraction.

Sometimes, the mnemonic "BEDMAS" is used, where "B" stands for "brackets," and serves the same purpose as "parentheses." The mnemonics essentially convey the same order of operations to reach the correct answer, but they use slightly different terminology based on regional preferences. For example, BEDMAS is more commonly used in Canada, while PEMDAS is prevalent in the U.S.

(Note that multiplication and division are of equal precedence in the order of operations, so the flipped order in BEDMAS doesn't change anything.)

The order of operations — as Americans know it today — was probably formalized in either the late 18th century. By the 20th century, the tool gained wider acceptance, coinciding with the rise of the U.S. textbook industry .

In an email, math and science historian Judith Grabiner explains that concepts like the order of operations are best thought of as "conventions, like red-means-stop and green-means-go, not mathematical truths.

"But once the convention is established," she says, "the analogy to traffic lights holds: Everybody's got to do it the same way and the 'same way' has to be 100 percent unambiguous."

Math and ambiguity are uncomfortable bedfellows.

PEMDAS ensures consistency in the results of mathematical calculations. Basically, when different people evaluate the same expression, they use the same process and come to the same result. If you don't follow the correct order of operations, you will likely get the wrong answer.

Ignoring or changing this order can lead to different results, which can be especially problematic in fields like science, engineering and finance where precise calculations are crucial.

Suppose it's finals week, and you're expected to solve the following equation:

Don't panic. This is where a certain auntie comes in. For every word in the phrase, "Please excuse my dear Aunt Sally," there's a corresponding math term (which begins with the same letter) that tells us which procedure(s) to perform first.

Parentheses First

Before we solve the equation, PEMDAS dictates that we ask ourselves a simple question: "Are there any parentheses?" If the answer is "yes," then our first move should be to resolve whatever's inside them.

So in the above example, we see " 2 x 3 " enclosed in the parentheses. Therefore, we'll begin by multiplying 2 times 3, which gives us 6. Now the equation looks like this:

Cool beans. Time to bring on the exponents ! In print, exponents take the form of a little number pressed against the upper righthand corner of a larger number. See the 5² ? That itty-bitty "2" is an exponent.

Here, the tiny two tells us to multiply 5 by itself. And 5 x 5 equals 25, giving us this:

Now that we've taken care of the parentheses and exponent(s), let's proceed to those next two operations: multiplication and division.

Multiply and Divide

Note that we're not saying multiplication comes before division here. Not necessarily, at least.

Let's say you're looking at a different problem which — at this stage — contains both a multiplication sign and a division symbol. Your job would be to perform the two operations in order from left to right .

The concept is best explained by way of example. If the equation reads 8 ÷ 4 x 3, first you'd divide the 8 by the 4, giving you 2. Then — and only then — would you multiply that 2 by 3. We now return to our regularly scheduled math problem:

Whoever wrote the original equation kept things nice and simple; there's nary a division sign in sight and only one multiplication symbol. Thank you, merciful exam gods.

Without further ado, we're going to multiply the 6 by the 4, giving us 24.

Time to Add and Subtract

As with multiplication and division, addition and subtraction are part of the same step. Once again, we're performing these two operations in order, from left to right. So we're going to have to subtract that 24 from the 9.

Doing so will give us a negative number, specifically -15.

But the 25 is a positive number. So in its current form, the equation consists of a negative 15 plus a positive 25. And when you add those two together, you get a positive 10.

So there it is. The answer to our riddle.

Double Parentheses: Proceed With Care

Before we part ways, there are a few more things you should know. You may someday find yourself looking at a complex equation with lots of different operations sandwiched between two parentheses. Maybe something like this:

Don't sweat it. If you're trying to solve math problems with multiple operations, following the PEMDAS sequence ensures consistent and accurate results. All you've got to do is work through the PEMDAS process inside those parentheses before you move on to the rest of the problem.

Here, you'd take care of the exponent first (i.e., the 2³), then handle the subtraction in that set of parenthesis before moving on to the multiplication in the next level of parenthesis. Easy-peasy. (In case you're interested, the answer to the equation is 2 1/3, or 2.33 if you prefer decimals.)

Here are some other PEMDAS-eque conventions and methods related to arithmetic expressions:

• BODMAS/BIDMAS : Used in the U.K. and other regions, BODMAS stands for B rackets, O rders (or I ndices for BIDMAS), D ivision, M ultiplication, A ddition and S ubtraction.
• FOIL : Specifically for binomials, it stands for F irst, O uter, I nner, L ast. It's a method for multiplying two binomials.
• Factorization : Breaking down numbers or expressions into their simplest components.
• Distributive property : For expressions like a(b + c), it would be ab + ac.
• Associative and commutative properties : Respectively, these properties allow numbers to be grouped differently or moved around in an expression without changing the result.
• Completing the square : A technique used in quadratic equations to convert them into a perfect square trinomial.
• Rationalizing the denominator : A method used to eliminate radicals from a denominator of a fraction.

This article was updated in conjunction with AI technology, then fact-checked and edited by a HowStuffWorks editor.

Robert Recorde — a physician and mathematician who was born in Wales in about 1510 C.E. — is credited as the inventor of the equal sign (=). He decided to use two parallel lines for this symbol because, in his words, "noe 2 thynges can be moare equalle [sic]."

Is PEMDAS wrong?

Why is pemdas in that order, what is the formula of pemdas, what is better bodmas or pemdas.

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Order of op

Order of operations

Here you will learn about order of operations , including what it means and how to calculate and solve order of operations problems using PEMDAS .

Students will first learn about order of operations as a part of operations and algebraic thinking in elementary school.

What is the order of operations?

Order of operations refers to the rule that explains the sequence of steps necessary for correctly evaluating a mathematical expression or math problem.

You will use the acronym PEMDAS to help recall the correct order, or priority, in which you complete mathematical operations.

Mathematical operations such as multiplication and addition have to be completed in a specific order. This sequence of steps helps us to evaluate any mathematical expression, both with numerical values and algebraic expressions.

To evaluate an expression using PEMDAS, you need to understand what PEMDAS represents and be able to apply the PEMDAS rule to any calculation.

PEMDAS stands for:

P arentheses, E xponents, M ultiplication and D ivision, A ddition and S ubtraction

Parentheses have a higher priority than exponents, so we calculate what is inside a pair of parentheses first. Exponents have a higher priority than division and multiplication, so any exponent that can be evaluated is calculated next, and so on.

The order can be remembered using the mnemonic device for PEMDAS, ‘Please Excuse My Dear Aunt Sally.’

It is important to note that multiplication and division are given equal priority , and addition and subtraction are given equal priority.

When completing calculations that involve multiplication and division or addition and subtraction, you work from left to right.

For example,

Consider the following expression 12-7+6.

12-7=5 and then 5+6=11

Consider the following expression 10 \div 5 \times 2.

10 \div 5=2 and then calculate 2 \times 2=4.

Visually you could represent PEMDAS as:

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

• Grade 5: Operations and Algebraic Thinking (5.OA.1) Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
• Grade 5: Operations and Algebraic Thinking (5.OA.2) Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
• Grade 6: Expressions and Equations (6.EE.1) Write and evaluate numerical expressions involving whole-number exponents.

How to evaluate using order of operations

In order to evaluate expressions using order of operations , you would use PEMDAS :

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

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Order of operations examples

Example 1: pemdas with addition and multiplication.

Evaluate 3+6\times{7}.

There are no parentheses.

2 Solve for any exponents.

There are no exponents.

3 Solve any division and multiplication calculations.

The multiplication that we need to calculate is 6\times{7}=42.

Replacing 6\times{7} with 42 gives us the calculation 3+42.

4 Solve any addition and subtraction calculations.

So 3+6\times{7}=45

Example 2: PEMDAS with division and subtraction

Evaluate 12-8\div{2}.

The division you need to calculate is 8\div{2}=4.

Replacing 8\div{2} with 4 gives us the calculation 12-4.

So 12-8\div{4}=8

Example 3: PEMDAS with parentheses and multiplication

Evaluate 3(2+4\times{6}-3).

You have a pair of parentheses and so you need to resolve what is inside the set of parentheses first. This is the calculation 2+4\times{6}-3.

Using PEMDAS, multiplication comes before addition and subtraction so you need to solve the multiplication 4\times{6}=24.

Replacing 4\times{6} with 24 in the calculation, you have 2+24-3.

Addition and subtraction should be completed from left to right and so you have 2+24=26 and 26-3=23. Therefore 2+24-3=23.

As 2+4\times{6}-3=23 and 3(2+4\times{6}-3) means 3\times(2+4\times{6}-3), you have the updated calculation 3\times{23}.

3\times{23}=69

There is nothing else to solve, so the final answer is:

3(2+4\times{6}-3)=69

Example 4: PEMDAS with an exponent and multiplication

Evaluate 4\times{3}^{2}.

Here you have to resolve 3^{2}=3\times{3}=9.

Replacing 3^{2} with 9 in the calculation, you now have:

The final step needed is to calculate 4\times{9}=36

4\times{3}^{2}=36

Example 5: PEMDAS with brackets, an exponent and division

Evaluate 3+(10\div{4}\times{20})^{2}.

Within the set of parentheses, you have the calculation 10\div{4}\times{20}.

You need to work from left to right.

Completing the division, you have 10\div{4}=2.5

Next, 2.5\times{20}=50

Therefore, 10\div{4}\times{20}=50.

Replacing 10\div{4}\times{20} with 50, you now have the updated calculation,

You now have to solve 50^{2}=50\times{50}=2500.

Updating the calculation, you now have 3+2500.

There are no divisions or multiplications to solve.

3+2500=2503 which gives us the final answer:

3+(10\div{4}\times{20})^{2}=2503

Example 6: PEMDAS with everything

Evaluate 4^{2}+2(14-8)\div{3}.

Within the brackets is the calculation 14-8=6.

Updating the calculation by changing the value in the bracket to 6, you have

4^{2}+2\times{6}\div{3}

As 4^2=4 \times 4=16, you now have

16+2\times{6}\div{3}

Here you have to calculate 2\times{6}\div{3}. Working from left to right, you calculate 2\times{6} and then divide the solution by 3.

2\times{6}=12

12\div{3}=4

Updating the calculation, you now have 16+4.

As 16+4=20, our final answer is:

4^{2}+2(14-8)\div{3}=20

Teaching tips for order of operations

• While PEMDAS is a great way for students to remember the order of operations, it’s important that the students understand how to solve for each of the operations required, not simply giving them the acronym to memorize.
• It is important to find a way for students to practice outside of order of operations worksheets. The more practice a student gets, the better. For example you can physically separate word problems from the answer choices in the classroom. Students would have to walk around to discuss and solve the problem and then find the answer that matches.
• Students could be given an expression without the symbols. They would have to then decide which operations/symbols would make the expression true.

Easy mistakes to make

• Multiplication/division and addition/subtraction in the incorrect order When you have a chain of multiplication and division calculations or a chain of addition and subtraction calculations, you work from left to right. For example, If you put parentheses into the calculation 12\div{4}\times{3} with the purpose of keeping the calculation exactly the same, you must place the parentheses around 12\div{4}. This is because 12\div{4}\times{3}=9, and (12\div{4})\times{3}=3\times{3}=9. If you placed parentheses around 4\times{3} you would have 12\div{(4\times{3})}=12\div{12}=1 which is a different answer that you would get without the parenthesis.
• Exponents as multiplying a number by 2 When introducing exponents, some students will interpret 4^2 as 4 \times 2 instead of 4 \times 4. The use of a number line to represent this expression or using expanded notation when introducing the concept can help students internalize the learning.
• Multiplying before exponents Take 2\times{3}^{4}. Because you would perform exponents before you perform multiplication within PEMDAS, you need to calculate the exponent term first, and then multiply the solution by 2. Inserting a pair of parentheses with the purpose of keeping the calculation exactly the same, 2\times{3}^{4}=2\times(3^{4})=162. A common misconception is that the multiplication is carried out first, and then the value is raised to the power; here this would look like (2\times{3})^{4} and is equal to 6^{4}=1296 which is the wrong answer.

Related properties of equality lessons

• Properties of equality
• Associative property
• Commutative property
• Distributive property

Practice order of operations questions

1. Calculate 6+5 \times 8.

To solve, follow the order of operations by using PEMDAS.

2. Calculate 16-15 \div 3.

To solve, follow the order of operations by using PEMDAS .

3. Calculate (8-2 \times 5+4) \div 2.

4. Calculate 2 \times 5^2.

5. Calculate 258-(10 \div 2 \times 3)^2.

6. Calculate 3(7-5)^3 \div 6+4.

Order of operations FAQs

In the order of operations, multiplication and division are seen as equal and would be performed as they are stated within the expression, starting from left to right. The same is true with addition and subtraction.

Both PEMDAS and BODMAS are correct. The acronym BODMAS is the UK version of the same rules. It can also be referred to as BIDMAS or BEDMAS . The acronym in the UK would read as: B rackets, I ndices, D ivision and M ultiplication, A ddition and S ubtraction.

An exponent refers to the number of times a number is multiplied by itself. For example 7^2 = 7 \times 7 or 4^3 = 4 \times 4 \times 4.

Parentheses are used within mathematical expressions to note a modification is the normal order of operations. In the expression, (3+4) \times 3, because the addition is within parentheses, it is solved before the multiplication.

The next lessons are

• Addition and subtraction
• Multiplication and division
• Types of numbers

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Order of Operations

Introduction.

The order of operations is a very simple concept, and is vital to correctly understanding math. Unlike reading, where we always work left-to-right, sometimes with math we need to work one part of a problem before another, or the final answer could be incorrect! We use the term "order of operations" to describe which part of the problem needs to be worked first. Take this equation as an example:

If you were to simply solve from left to right, the answer would be incorrect. Let's do that now: 4 + 6 = 10. Divide that by 2 to get 5. Multiply 5 times 11 to get 55. Unfortunately, even though it seemed ok, this answer is wrong.

The correct order of operations

The order of operations will allow you to solve this problem the right way. The order is this: Parenthesis , Exponents , Multiplication and Division , and finally Addition and Subtraction . Always perform the operations inside a parenthesis first, then do exponents. After that, do all the multiplication and division from left to right, and lastly do all the addition and subtraction from left to right.

A popular way of remembering the order is the acronym PEMDAS. Parenthesis, Exponents, Multiply and Divide, Add and Subtract. You can also create a little phrase to go along with this, like " P lease E xcuse M y D ear A unt S ally." Whatever you choose, make sure that you know all six steps of the order of operations very well.

Let's try solving that equation again, this time using PEMDAS.

Step 1) Parenthesis. There aren't any. Move on.

Step 2) Exponents. None. Keep going...

Step 3) Multiplication and Division. Go from left to right performing all the multiplication and division as you come across it, so divide 6 by 2 to get 3, and multiply that by 11 to get 33.

Step 4) Addition and Subtraction. From left to right, 4 + 33 = 37.

The whole idea is just to follow the rule: PEMDAS. Now we can try to solve one with parenthesis and exponents.

What if you are given an equation like the one below? Just simplify it in small steps, using the order of operations at all times.

Remember, the first step is Parenthesis. Look inside the parenthesis, where we have \(6*2\). That comes out to 12. We can then compute \(12^2\) and get 144, making this equation easy to finish:

By now you should have a basic understanding of the order of operations. To continue learning more about this subject, you can keep browsing our site, or try a web search on Yahoo or Google . MathGoodies.com also offers a great lesson on the Order of Operations .

Order of Operations

It seems that each student interpreted the problem differently, resulting in two different answers. Student 1 performed the operation of addition first, then multiplication; whereas student 2 performed multiplication first, then addition. When performing  arithmetic operations  there can be only one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation.

Rule 1: First perform any calculations inside parentheses.

Rule 2: Next perform all multiplications and divisions, working from left to right.

Rule 3: Lastly, perform all additions and subtractions, working from left to right.

An easy way to remember this order is to use the acronym  PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction).

The above problem was solved correctly by Student 2, since she followed Rules 2 and 3. Let’s look at some examples of solving arithmetic expressions using these rules.

Example 1: Evaluate each expression using the rules for PEMDAS.

In Example 1, each problem involved only 2 operations. Let’s look at some examples that involve more than two operations.

Example 2: Evaluate 3 + 6 x (5 + 4)  ÷  3 – 7 using PEMDAS.

Example 3: Evaluate 9 – 5  ÷  (8 – 3) x 2 + 6 using PEMDAS.

In Examples 2 and 3, you will notice that multiplication and division were evaluated from left to right according to Rule 2. Similarly, addition and subtraction were evaluated from left to right, according to Rule 3.

When two or more operations occur inside a set of parentheses, these operations should be evaluated according to Rules 2 and 3. This is done in Example 4 below.

Example 4: Evaluate 150  ÷  (6 + 3 x 8) – 5 using the PEMDAS.

Solution: 

Example 5: Evaluate the arithmetic expression below:

Solution: This problem includes a  fraction bar  (also called a vinculum), which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing.

Evaluating this expression, we get:

Example 6: Write an arithmetic expression for this problem. Then evaluate the expression using PEMDAS.

Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him?

Solution: 32 + 3 x 15   =   32 +  3 x 15     =   32 + 45 =   77

Jill owes Mr. Smith $77.

Summary: When evaluating arithmetic expressions, the operation order is:

• Simplify all operations inside parentheses.
• Perform all multiplications and divisions, working from left to right.
• Perform all additions and subtractions, working from left to right.

If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator.

Directions: Complete each exercise by applying the rules for order of operations. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

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 .

Order of Operations Lessons

The Order of Operations is very important when simplifying expressions and equations. The Order of Operations is a standard that defines the order in which you should simplify different operations such as addition, subtraction, multiplication and division.

This standard is critical to simplifying and solving different algebra problems. Without it, two different people may interpret an equation or expression in different ways and come up with different answers. The Order of Operations is shown below.

• Parentheses and Brackets — Simplify the inside of parentheses and brackets before you deal with the exponent (if any) of the set of parentheses or remove the parentheses.
• Exponents — Simplify the exponent of a number or of a set of parentheses before you multiply, divide, add, or subtract it.
• Multiplication and Division — Simplify multiplication and division in the order that they appear from left to right.
• Addition and Subtraction — Simplify addition and subtraction in the order that they appear from left to right.

Before we begin simplifying problems using the Order of Operations, let’s examine how failure to use the Order of Operations can result in a wrong answer to a problem.

Without the Order of Operations one might decide to simplify the problem working left to right. He or she would add two and five to get seven, then multiply seven by x to get a final answer of 7x. Another person might decide to make the problem a little easier by multiplying first. He or she would have first multiplied 5 by x to get 5x and then found that you can’t add 2 and 5x so his or her final answer would be 2 + 5x. Without a standard like the Order of Operations, a problem can be interpreted many different ways.

Order of Operations Example

Below is the first expression that we will be simplifying:

Order of Operations

• Parentheses and Brackets from the inside out
• Exponents of numbers or parentheses
• Multiplication and Division in the order they appear.
• Addition and Subtraction in the order they appear.

As we walk through the steps to simplifying this expression, use the Order of Operations reference in the right column of this page. The first step in the Order of Operations is to simplify parentheses and brackets from the inside out. You must remember to use the Order of Operations when simplifying the inside of the parentheses, but we don’t need to worry about that in this problem because there is only one operation inside the parentheses 3 – 1. In this case all that has to be done is subtraction of 1 and 3. The answer is shown below.

The next step in the Order of Operations is to simplify exponents. 3 2 becomes 9. The result is shown below.

The next step in the Order of Operations is to simplify multiplication and division in the order that they appear. There is no division, only multiplication. Multiply (2) and 9:

The final step is to simplify addition and subtraction (combine like terms).

Order of Operations Example (2)

Below is the next expression that we will be simplifying:

Again, the first step in the Order of Operations is to simplify parentehsis and brackets from the inside out. The polynomial x + 1 is in the innermost set of parentheses, but nothing inside of it can be simplified. Now we can simplify the second innermost grouping symbol, the bracket using the Order of Operations.

First simplify multiplication and division in the order they appear. There is no division, all we need to do is distribute 8 into (x + 1).

Now remove the parentheses symbols and simplify addition and subtraction in the order that they appear (combine like terms).

Now, start using the Order of Operations to simplify the polynomial inside of the second set of parentheses. There are no exponents on the inside so you can skip to simplifying multiplication and division in the order the appear. The division comes first in this expression, so divide 3 by 2 first.

Multiplication comes second in this problem, so now you can multiply three halves by 2.

Now that the inside of each set of parentheses and brackets are simplified, the problem can be worked on as a whole instead of in little groups. Now in step two of the Order of Operations, simplify the only exponent in the expression, (3) 2

Continuing with the Order of Operations, multiply [8x + 10] and (9).

There are no terms which can be combined, this problem is complete.

Examples of the Order of Operations

Evaluate the following

First, evaluate whatever is in the parentheses:

Next, evaluate the exponent:

Evaluate any multiplication and division from left to right:

Evaluate any addition and subtraction from left to right or which ever way makes it easier for you:

is evaluated as

Solve for x in the equation below

As always evaluate the expression within the parentheses first, since there is more than one operator in the parentheses, apply PEMDAS to the expression

Divide both sides by 21

note that in the above expression, one can choose to divide first since that makes the computation easier

Quiz on the Order of Operations

Step 1: Evaluate the inner parentheses first

Step 2: Evaluate the multiplication within the parentheses

Step 3: Evaluate the subtraction within the parentheses

Step 4: Finally evaluate the last multiplication

Order of Operations Resources

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The Order of Operations in Math — A Complete & Kid-Friendly Guide

Whether you're getting ready for an exam or simply want to refresh your knowledge of the order of operations in math, we've created a comprehensive and easy-to-follow guide for you.

Read on to find simple definitions of the order of operations and related terms, practical examples, and a fun quiz to test your knowledge.

What Is the Order of Operations?

Have you ever tried to solve a math problem, where you didn't know where to start or what to do next?

For example, when you see 3 + 8 × 2 - 6, what should you do first?

• Add 3 and 8
• Multiply 8 and 2
• Subtract 6 from 2

The order of operations is a set of rules that tells us what math operation to do first in an expression with multiple operations like addition, subtraction, multiplication, and division.

Following the order of operations, when solving 3 + 8 × 2 - 6, we would first do the:

• Multiplication: 8 x 2 = 16, so we get 3 + 16 – 6.
• Addition: 3 + 16 = 19, so we get 19 – 6.
• Subtraction: 19 – 6 = 13.

3 + 8 × 2 – 6 = 13

One way to remember the order of operations is by the acronym PEMDAS.

Watch our video to learn more.

What Does “PEMDAS” Mean?

PEMDAS is an acronym that reminds us of the sequence of steps to follow when solving mathematical expressions with multiple operations.

• P  stands for Parentheses (sometimes called Brackets).
• E  stands for Exponents (sometimes called Indices).
• M stands for Multiplication.
• D  stands for Division.
• A stands for Addition.
• S  stands for Subtraction.

Much like grammatical rules that govern how we build sentences, PEMDAS is a set of rules mathematicians came up with and agree to follow so that we can have consistency and clarity when performing calculations.

Different countries have their own ways of remembering the order of math operations.

• In Canada, it's called BEDMAS : Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
• Some prefer BODMAS : Brackets, Order, Division, Multiplication, Addition, and Subtraction.
• Then there's GEMDAS : Grouping, Exponents, Multiplication, Division, Addition, and Subtraction. 

When solving math expressions with multiple operations, we follow these steps from left to right .

There is a fun way of remembering the PEMDAS rule with the left to right rule using a playful mnemonic:

Now, let’s work through each step of PEMDAS.

1. Parentheses

The first step to the order of operations is parentheses. This means that when we’re solving an expression with multiple operations, we first solve the operations within parentheses (). Here’s a simple example to illustrate.

3 x (4 + 2)

• First, we solve the expression within the parentheses: 4 + 2 =6, so we get 3 x 6.
• Then, we multiply 3 by the result of the expression inside the parentheses (which, as we've seen, is 6): 3 x 6.
• Finally, we get the answer 18.  

Sometimes, math expressions contain a set of parentheses inside another set. To solve them, start by solving the operations inside the innermost parentheses , then work your way outward.

For example: {4 x [5 +(6 – 3)]} – (8 – 2)

• First, we subtract numbers inside the innermost parentheses: 6 – 2 = 3, so we get {4 x [5 + 3]} – (8 – 2).
• Then, add 5 to the answer inside the square brackets: 5 + 3 = 8, so we get {4 x 8} – (8 – 2). 
• Next, multiply the result by 4 inside the curly brackets: 4 x 8 = 32, so we get 32 — (8 – 2). 
• Now, solve the expression inside the parantheses: 8 – 2 = 6, so we get 32 – 6. 
• Finally, subtract the result of step 4 from the result of step 3: 32 – 6 = 26

2. Exponents

The second step in the order of operations is exponents. Exponents, also known as indices, show how many times a number is multiplied by itself.

To illustrate, 3⁴ is an exponent form that means multiply 3 by itself 4 times, or 3 x 3 x 3 x 3.

Let’s look at this example:

2 2 + 4 x 2

• First, we look at the exponent 2 3 which means we multiply 2 by itself 3 times, or 2 × 2 × 2 = 8 , so we get: 8 + 4 × 2. 
• Then we multiply: 4 × 2 = 8 , so we get 8 + 8. 
• Finally, we add: 8 + 8 = 16.

Refresh Your Memory:  What Does Squared Mean in Math?

 3. Multiplication & Division

In the order of operations, multiplication and division are prioritized equally and are performed from left to right as they appear in the expression. This is why we consider them together as the third step in PEMDAS.

Let’s see that in practice with: 8 ÷ 2 × 4 + 3

• First, we perform the division because it appears first, before the multiplication: 8 ÷ 2 = 4, so we get 4 x 4 + 3.
• Then, we perform the multiplication: 4 x 4 = 16, so we get 16 + 3.
• Next, we perform the addition: 16 + 3 = 19.
• Finally, we get the result 19.

4. Addition & Subtraction 

As the last steps of PEDMAS, we work on the addition and subtraction.

Just like multiplication and division, addition and subtraction are prioritized equally and are performed from left to right as they appear in the expression.

For example:

10 – 4 + 3 

• First, we subtract: 10 – 4 = 6, so we get 6 + 3
• Then, we add: 6 + 3 = 9
• Thus, the result is 9.

Now that we have covered each step of PEMDAS, let’s look at a more challenging example:

Find more tips in our Order of Operations Series Part 2 .

Common Mistakes with the Order of Operations in Math 

PEMDAS makes solving math expressions much easier than simply memorizing the order of operations, but students sometimes forget the steps to take.

Here are some common mistakes to keep an eye on when solving math expressions with multiple operations:

1. Going Only Left to Right

Some students forget PEMDAS and just focus on the left-to-right approach, which might feel natural because of how we read.

It's important to remember that while going left to right can work, it's only suitable when you're solving expressions that only contain addition and subtraction, or multiplication and division.

For example, using the left to right approach works if you are trying to solve:

8 – 6 + 9 – 3

Or, if you are trying to solve:

8 ÷ 2 × 9 ÷ 3

But let's say you are trying to solve:

10 – 3 x 2 + 5 ÷ 5

As you can see, this expression contains subtraction, multiplication, addition, and division.

If we go from left to right – first subtracting 3 from 10 to get 7, then multiplying 7 by 2 to get 14, then adding 5 to it to get 19, and finally diving 19 by 5, we get 3.8 which is an incorrect result.

But if we follow PEMDAS, which tells us to start with multiplication and division (left to right) before we get to addition and subtraction, we get:

= 10 – 6 + 1 

  The correct answer is 5 .

2. Doing Addition Before Subtraction

Many students tend to do addition before subtraction in math problems because they learned addition before subtraction or because they're close together in the order of operations. Some also think that because A comes before S in PEMDAS you should always do addition before subtraction.

For instance, in the expression  8 – 2 + 3  you might tackle the addition before the subtraction and get: 8 –  5 = 3.

In the order of operations, addition and subtraction have the same priority which is why we have to use left-to-right approach and perform subtraction first:

= 6 + 3 

3. Doing Multiplication Before Division

Similarly to addition vs. subtractions, students often think multiplication comes before division because that's what they've learned before or because they're used to seeing multiplication listed first in math problems. Another mistake is believing M comes before D in PEMDAS because you always do multiplication before division. 

For example, in the expression 6 ÷ 2 × 3 = 6 ÷ 6 = 1 , a student might do the multiplication first and get a wrong result:

6 ÷ 2 × 3 = 6 ÷ 6 = 1 

But we know by now that multiplication and division are equally important. So, we need to work from left to right and do the division first:

6 ÷ 2 × 3 = 3 × 3 = 9

Solved Examples for the Order of Operations

Using PEMDAS, let’s solve this one: 

3 x (8 – 4) 2 + 6 ÷ 2

• First, we do the subtraction inside the parentheses: 8 – 4 = 4, so we get 3 x 4 2  + 6 ÷ 2.
• Next, we square the result: 4 2  = 16, so we get 3 x 16 + 6 ÷ 2.
• Then, we multiply: 3 x 16 = 48, so we get 48 + 6 ÷ 2.
• Next, we divide: 6 ÷ 2 = 3, so we get 48 + 3.
• Finally, we add: 48 + 3 = 51.

Example 2 

Let’s do the same for this expression:

3 x (7 – 2) + 6 ÷ 2

• First, we do the subtraction inside the parentheses: 7 – 2 = 5, so we get 3 x 5 +  6 ÷ 2.
• Next, we multiply: 3 x 5 = 15, so we get 15 +  6 ÷ 2.
• Then, we divide: so we get 6 ÷ 2 = 3, so we get 15 + 3.
• Finally, we add:15 + 3 = 18.

Finally, we can work on this one:

80  ÷ (6 + 7 x 2) – 5

• First, we perform the multiplication inside the parentheses: 7 x 2 = 14, so we get 80 ÷ (6 + 14) – 5.
• Next, we perform the addition inside the parentheses: 6 + 14 = 20, so we get 80 ÷ 20 – 5.
• Then, we divide: 80 ÷ 20 = 4, so we get 4 – 5.
• Finally, we subtract and get: 4 – 5 = –1.

Flash Quiz: How Well Do You Know the Order of Operations?

Frequently asked questions about the order of operations.

Discover clear and easy-to-understand explanations about the rules and steps for solving math problems using the order of operations.

1. Can PEMDAS be applied to all math operations?

Yes, PEMDAS applies to all mathematical operations involving multiple steps, including addition, subtraction, multiplication, division, and exponentiation

2. What are BODMAS and BEDMAS?

BODMAS  and BEDMAS , just like PEDMAS  are mnemonic devices used to remember the order of operations in mathematics:

• B: Brackets
• D: Division
• M: Multiplication
• A: Addition
• S: Subtraction
• E: Exponents

3. What should I do when math expressions have parentheses within parentheses?

When there are parentheses within one another (nested parentheses), you should start by doing the innermost expressions first and then work your way outwards.

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Algebra Topics  - Order of Operations

Algebra topics  -, order of operations, algebra topics order of operations.

Algebra Topics: Order of Operations

Lesson 1: order of operations, introduction to the order of operations.

How would you solve this problem?

12 - 2 ⋅ 5 + 1

The answer you get will depend largely on the order in which you solve the problem. For example, if you work the problem from left to right —12-2, then 10⋅5, then add 1—you'll get 51 .

12 - 2 ⋅ 5 + 1 10 ⋅ 5 + 1 50 + 1 51

On the other hand, if you solve the problem in the opposite direction—from right to left —the answer will be 0 .

12 - 2 ⋅ 5 + 1 12 - 2 ⋅ 6 12 - 12 0

Finally, what if you did the math in a slightly different order? If you multiply first, then add , the answer is 3 .

12 - 2 ⋅ 5 + 1 12 - 10 + 1 2 + 1 3

It turns out that 3 actually is the correct answer because it's the answer you get when you follow the standard order of operations . The order of operations is a rule that tells you the right order in which to solve different parts of a math problem. ( Operation is just another way of saying calculation. Subtraction, multiplication, and division are all examples of operations.)

The order of operations is important because it guarantees that people can all read and solve a problem in the same way. Without a standard order of operations, formulas for real-world calculations in finance and science would be pretty useless—and it would be difficult to know if you were getting the right answer on a math test!

Using the order of operations

The standard order of operations is:

Parentheses

Multiplication and division, addition and subtraction.

In other words, in any math problem you must start by calculating the parentheses first, then the exponents , then multiplication and division , then addition and subtraction . For operations on the same level, solve from left to right . For instance, if your problem contains more than one exponent, you'd solve the leftmost one first, then work right.

Let's look at the order of operations more closely and try another problem. This one might look complicated, but it's mainly simple arithmetic. You can solve it using the order of operations and some skills you already have.

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2) + 18 / 3 2 - 8

Always start with operations contained within parentheses. Parentheses are used to group parts of an expression.

If there is more than one set of parentheses, first solve for the ones on the left. In this problem, we only have one set:

In any parentheses, you follow the order of operations just like you do with any other part of a math problem.

Here, we have two operations: addition and multiplication . Because multiplication always comes first, we'll start by multiplying 6 ⋅ 2 .

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2 ) + 18 / 3 2 - 8

6 ⋅2 is 12. Next, we'll add 4 .

4 / 2 ⋅ 3 + ( 4 + 12 ) + 18 / 3 2 - 8

4+12 is 16 . So we've simplified our parentheses to 16 . Since we just have a single number in the parentheses, we can get rid of them all together—they're not grouping together anything now.

4 / 2 ⋅ 3 + 16 + 18 / 3 2 - 8

Second, solve any exponents . Exponents are a way of multiplying a number by itself. For instance, 2 3 is 2 multiplied by itself three times, so you would solve it by multiplying 2 ⋅2 ⋅2 . (To learn more about exponents, review our lesson here ).

There's only one exponent in this problem : 3 2 . 3 2 is 3 multiplied by itself twice —in other words, 3 ⋅ 3 .

3 ⋅ 3 is 9 , so 3 2 can be simplified as 9 .

4 / 2 ⋅ 3 + 16 + 18 / 9 - 8

Next, look for any multiplication or division operations. Remember, multiplication doesn't necessarily come before division—instead, these operations are solved from left to right .

Starting from the left means that we need to solve 4 / 2 first.

4 divided by 2 is 2 . That makes our next problem 2 ⋅ 3 .

2 ⋅ 3 + 16 + 18 / 9 - 8

2 ⋅ 3 is 6 . Finally, there's only one multiplication or division problem left: 18 / 9 .

6 + 16 + 18 / 9 - 8

18 / 9 is 2 . There's nothing left to multiply or divide, so we can move on to the next and final part of the Order of Operations: addition and subtraction .

6 + 16 + 2 - 8

Our problem looks a lot simpler to solve now. All that's left is addition and subtraction.

Just like we did with multiplication and division, we'll add and subtract from left to right . That means that first we'll add 6 and 16 .

6 + 16 is 22 . Next, we need to add 22 to 2 .

22 + 2 is 24 . Only one operation left: 24 - 8 .

24-8 is 16 . That's it!

We're done! We've solved the entire problem, and the answer is 16 . In other words, 4 / 2 ⋅ 3 + ( 4 + 6 ⋅ 2 ) + 18 / 3 2 - 8 equals 16 .

4 / 2 ⋅ 3 + (4 + 6 ⋅ 2) + 18 / 3 2 - 8 = 16

Whew! That was a lot to say, but once we broke it down into the right order it really wasn't that complicated to solve. When you're first learning the order of operations, it might take you a while to solve a problem like this. With enough practice, though, you'll get used to solving problems in the right order.

Remembering the order of operations

If you use it a lot, you'll eventually get the hang of the order of operations. Until then, it can be helpful to use a word or phrase to remember it. Two popular ones are the nonsense word PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and the phrase Please Excuse My Dear Aunt Sally .

/en/algebra-topics/exponents/content/

Order of Operations (PEMDAS)

Imagine you’re getting ready for a walk. You grab your shoes and socks and head to the couch. Which do you put on first: shoes or socks? Of course, it’s your socks. That’s the correct order.

Now, imagine two friends attempting to simplify this expression: \(28 – 3 × 5 + 10\) . The first friend simplifies the expression to 23, and the second friend simplifies the expression to 135. Who is right? The first friend. They remembered to use the order of operations. The second friend “tried to put the shoes on first,” meaning they computed the first operation they came across, giving them an incorrect result. Just like prepping for your walk, simplifying expressions requires a set order, also known as the order of operations.

PEMDAS Meaning

Many people use the acronym PEMDAS to help remember the order of operations. PEMDAS stands for “Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.” The acronym also helps to remember that we start simplifying the expression from left to right, just like we read “PEMDAS” from left to right.

PEMDAS Examples

Let’s use PEMDAS to take a closer look at the order of operations.

\(28 – 3 × 5 + 10\)

Start by scanning the expression for “P” – parentheses. If you encounter a multiple-term expression inside the parentheses, continue to apply the same order of operations inside the parentheses to accurately simplify it before moving on to the next step. It’s worth noting that you should also look for brackets and braces in this step as well.

There are no parentheses in this expression, so we continue to the next operation: “E” for exponents.

Exponents can be found anywhere in the expression, including on the outside of parentheses. When this happens, be sure to compute what’s inside the parentheses before applying the exponent. There are no exponents in this expression, so we move on to the next step.

Now, we are looking for “MD” (either multiplication or division). Multiplication and division are inverse operations, so they are considered a set. This means we would use whichever operation comes first, from left to right.

This expression has a multiplication operator ( \(3 × 5\) ), so we compute that first: \(3 × 5 = 15\) . To stay organized, we rewrite the expression below the original, showing our result for the first operation, \(28 – 3 × 5 + 10\) becomes \(28 – 15 + 10\) .

We are ready to look for the last set of operations and finish simplifying this expression. The final set of operations “AS” – addition and subtraction. Just like multiplication and division, addition and subtraction are a set of inverse operations, so we treat them the same by using whichever operation comes first from left to right. In this expression, subtraction comes first, so we compute the subtraction, \(28 – 15 = 13\) , and then compute the addition: \(13 + 10 = 23\) .

Okay, before we work through some other problems, let’s review what PEMDAS stands for: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Follow this order of operations and complete any inverse operations (multiplication/division and addition/subtraction) from left to right.

All right, let’s use PEMDAS to solve the following practice questions:

1. Use the order of operations to simplify this expression to three terms.

\((73 – 5^2 × 2) + 21 – 3\)

• \(5^2 × 2 +18\)
• \(73 – 50 +18\)
• \(23 + 21 -3\)
• \(48 + 21 -3\)

Show Answer The correct answer is C!

The first step when simplifying this expression is the exponent inside the parentheses.

\((73 – 25 × 2) + 21 – 3\)  

Next is the multiplication inside the parentheses.

\((73 – 25 × 2) + 21 – 3\)

\((73 – 50) + 21 – 3\)  

Finally, subtract inside the parentheses to simplify to three terms.

\((73 – 50) + 21 – 3\)

\(23 + 21 – 3\)  

If you selected choice A or B, you subtracted too soon and did not compute inside the parentheses first. If you selected choice D, you did not use the order of operations inside the parentheses. The first step to simplifying this expression is to compute the exponent inside the parentheses, then multiply, and then subtract.

2. Look at the two simplifications for the expression.

Which student is correct, and what mistake did the other student make?

• Amir, Bart did not add before subtracting.
• Amir, Bart did not subtract before adding
• Bart, Amir did not add before subtracting.
• Bart, Amir did not subtract before adding.

Show Answer The correct answer is B!

When computing addition and subtraction in an expression, always work from left to right and use the operation to the left first. In this expression, that is subtraction. So, Amir’s simplification is correct, and Bart added before subtracting.

That’s all for this review. Thanks for watching, and happy studying!

Frequently Asked Questions

What is the order of operations in math.

The order of operations is the order you use to work out math expressions: parentheses, exponents, multiplication, division, addition, subtraction. All expressions should be simplified in this order. The only exception is that multiplication and division can be worked at the same time, you are allowed to divide before you multiply, and the same goes for addition and subtraction. However, multiplication and division MUST come before addition and subtraction. The acronym PEMDAS is often used to remember this order.

Ex. Use the order of operations to simplify the expression \(3×4^2+8-(11+4)^2÷3\).

Parentheses: \(3×4^2+8-(15)^2÷3\)

Exponents: \(3×16+8-225÷3\)

Multiplication/Division: \(48+8-75\)

Addition/Subtraction: \(-19\)

Do you use the order of operations when there are no parentheses?

Yes, always use the order of operations to simplify expressions. If there are no parentheses, then skip that step and move on to the next one. The same applies for any other missing operation.

Ex.Use the order of operations to simplify the expression \(6^2-4+2\).

Parentheses: There are none, so skip this step.

Exponents: \(36-4+2\)

Multiplication/Division: There isn’t any, so skip this step.

Addition/Subtraction: \(34\)

Do calculators do order of operations?

No, most calculators do not follow the order of operations, so be very careful how you plug numbers in! Make sure you follow the order of operations, even if that means plugging in numbers in a different order from how they look on your page.

Which math operation comes first?

Parentheses are the first operation to solve in an equation. If there are no parentheses, then move through the order of operations (PEMDAS) until you find an operation you do have and start there.

What are the basic operations?

The four basic operations are: addition (+), subtraction (-), multiplication (×), and division (÷).

Math Order of Operations Rules PDF

Order of Operations Practice Problems

  \(7\times9+3-6\div2+2^2-11\)

The correct answer is 56. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. \(7×9+3-6÷2+2^2-11\) There are no parentheses in this problem, so start with exponents. \(7×9+3-6÷2+4-11\) Then, multiply and divide from left to right. \(63+3-6÷2+4-11\) \(63+3-3+4-11\) Finally, add and subtract from left to right. \(66-3+4-11\) \(63+4-11\) \(67-11\) \(56\)

  \(19+7(26-48÷2)^3+3×6\)

The correct answer is 93. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. \(19+7(26-48÷2)^3+3×6\) First, start with parentheses. The order of operations must be followed even inside parentheses, so be sure to divide before you subtract. \(19+7(26-24)^3+3×6\) \(19+7(2)^3+3×6\) Next comes exponents. \(19+7(8)+3×6\) Then, multiply from left to right. \(19+56+3×6\) \(19+56+18\) Finally, add from left to right. \(75+18\) \(93\)

  \(11+3-7×2+1×4÷2\)

The correct answer is 2. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. \(11+3-7×2+1×4÷2\) There are no parentheses or exponents, so start with multiplication and division from left to right. \(11+3-14+1×4÷2\) \(11+3-14+4÷2\) \(11+3-14+2\) Finally, add and subtract from left to right. \(14-14+2\) \(0+2\) \(2\)

  \(3(11+2)^2-18÷6\)

The correct answer is 504. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. \(3(11+2)^2-18÷6\) First, simplify what is in parentheses. \(3(13)^2-18÷6\) Then, do any exponents. \(3(169)-18÷6\) Next, multiply and divide from left to right. \(507-18÷6\) \(507-3\) Finally, subtract. \(504\)

  \((16-24)^2+3×11-1\)

The correct answer is 96. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. \((16-24)^2+3×11-1\) First, simplify the parentheses. \((-8)^2+3×11-1\) Then, do exponents. \(64+3×11-1\) Next, multiply. \(64+33-1\) Finally, add and subtract from left to right. \(97-1\) \(96\)

Order of Operations Worksheet

Use our free printable order of operations worksheets for additional practice!

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by Mometrix Test Preparation | Last Updated: August 30, 2024

Order of Operations Worksheets

Welcome to the order of operations worksheets page at Math-Drills.com where we definitely follow orders! This page includes Order of Operations worksheets using whole numbers, integers, decimals and fractions.

Elementary and middle school students generally use the acronyms PEMDAS or BEDMAS to help them remember the order in which they complete multi-operation questions. The 'P' or 'B' in the acronym stands for parentheses or brackets. All operations within parentheses get completed first. The 'E' refers to exponents; all exponents are calculated after the parentheses. The 'M' and 'D' are interchangeable as one completes the multiplication and division in the order that they appear from left to right. The fourth and final step is to solve for the addition and subtraction in the order that they appear from left to right.

More recently, students are being taught the acronym, PEMA, for order of operations, to avoid the confusion inherent in the other acronyms. For example, in PEMDAS, multiplication comes before division which some people incorrectly assumes means that multiplication must be done before division in an order of operations question. In fact, the two operations are completed in the order that they occur from left to right in the question. This is recognized in PEMA which more correctly shows that there are four levels to complete in an order of operations question.

Unless you want your students doing something different than the rest of the world, it would be a good idea to get them to understand these rules. There is no discovery or exploration needed here. These are rules that need to be learned and practiced and have been accepted as the standard approach to solving any multi-step mathematics problem.

Most Popular Order of Operations Worksheets this Week

Order of Operations With Whole Numbers and Integers

The worksheets in this section include questions with parentheses, addition, and multiplication. Exponents, subtraction, and division are excluded. The purpose of excluding some parts of PEMDAS is to ease students into how the order of operations works. To help students see a purpose for the order of operations, try to associate the expressions with related scenarios. For example, 2 + 7 × 3 could refer to the number of days in two days and three weeks. (9 + 2) × 15 could mean the total amount earned if someone worked 9 hours yesterday and 2 hours today for $15 an hour.

• Order of Operations With Whole Numbers (Addition and Multiplication Only) 2-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 3-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 4-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 5-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only ) 6-Step Order of Operations with Whole Numbers ( Addition & Multiplication Only )

The worksheets in this section include questions with parentheses, addition, subtraction, and multiplication. Exponents and division are excluded. This section is similar to the previous one in that it is meant to help ease students into the order of operations without complicating things with exponents and division.

• Order of Operations With Whole Numbers (Addition, Subtraction and Multiplication Only) 2-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 3-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 4-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 5-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only ) 6-Step Order of Operations with Whole Numbers ( Addition, Subtraction & Multiplication Only )

One last section to help ease students into the order of operations or simply for students who haven't learned about exponents yet. The questions on the worksheets in this section include parentheses and all four operations.

• Order of Operations With Whole Numbers (No Exponents) 2-Step Order of Operations with Whole Numbers ( No Exponents ) 3-Step Order of Operations with Whole Numbers ( No Exponents ) 4-Step Order of Operations with Whole Numbers ( No Exponents ) 5-Step Order of Operations with Whole Numbers ( No Exponents ) 6-Step Order of Operations with Whole Numbers ( No Exponents )

The worksheets in this section include questions with parentheses, exponents and all four operations.

• Order of Operations With Whole Numbers (All Operations, Parentheses and Exponents) 2-Step Order of Operations with Whole Numbers 3-Step Order of Operations with Whole Numbers 4-Step Order of Operations with Whole Numbers 5-Step Order of Operations with Whole Numbers 6-Step Order of Operations with Whole Numbers
• Order of Operations With Integers (No Exponents) 2-Step Order of Operations with Integers and No Exponents 3-Step Order of Operations with Integers and No Exponents 4-Step Order of Operations with Integers and No Exponents 5-Step Order of Operations with Integers and No Exponents 6-Step Order of Operations with Integers and No Exponents

The worksheets in this section include parentheses, exponents, and all four operations.

• Order of Operations With Integers (All Operations, Parentheses and Exponents) 2-Step Order of Operations with Integers 3-Step Order of Operations with Integers 4-Step Order of Operations with Integers 5-Step Order of Operations with Integers 6-Step Order of Operations with Integers

Order of Operations With Fractions and Decimals

As with other order of operation worksheets, the fractions order of operations worksheets require some prerequisite knowledge. If your students struggle with these questions, it probably has more to do with their ability to work with fractions than the questions themselves. Observe closely and try to pin point exactly what prerequisite knowledge is missing then spend some time going over those concepts/skills before proceeding. Otherwise, the worksheets below should have fairly straight-forward answers and shouldn't result in too much hair loss.

• Order of Operations with Positive Fractions (No Exponents) 2-Step Order of Operations with Positive Fractions (No Exponents) 3-Step Order of Operations with Positive Fractions (No Exponents) 4-Step Order of Operations with Positive Fractions (No Exponents) 5-Step Order of Operations with Positive Fractions (No Exponents) 6-Step Order of Operations with Positive Fractions (No Exponents)
• Order of Operations with Positive Fractions 2-Step Order of Operations with Positive Fractions 3-Step Order of Operations with Positive Fractions 4-Step Order of Operations with Positive Fractions 5-Step Order of Operations with Positive Fractions 6-Step Order of Operations with Positive Fractions
• Order of Operations with Positive and Negative Fractions 2-Step Order of Operations with Positive & Negative Fractions 3-Step Order of Operations with Positive & Negative Fractions 4-Step Order of Operations with Positive & Negative Fractions 5-Step Order of Operations with Positive & Negative Fractions 6-Step Order of Operations with Positive & Negative Fractions
• Order of Operations With Positive Decimals 2-Step Order of Operations with Positive Decimals 3-Step Order of Operations with Positive Decimals 4-Step Order of Operations with Positive Decimals 5-Step Order of Operations with Positive Decimals 6-Step Order of Operations with Positive Decimals
• Order of Operations With Positive and Negative Decimals 2-Step Order of Operations with Positive & Negative Decimals 3-Step Order of Operations with Positive & Negative Decimals 4-Step Order of Operations with Positive & Negative Decimals 5-Step Order of Operations with Positive & Negative Decimals 6-Step Order of Operations with Positive & Negative Decimals
• Order of Operations With Positive Decimals (European Format: Comma Decimal) 2-Step Order of Operations with Positive Decimals (Comma Decimal) 3-Step Order of Operations with Positive Decimals (Comma Decimal) 4-Step Order of Operations with Positive Decimals (Comma Decimal) 5-Step Order of Operations with Positive Decimals (Comma Decimal) 6-Step Order of Operations with Positive Decimals (Comma Decimal)
• Order of Operations With Positive and Negative Decimals (European Format: Comma Decimal) 2-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 3-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 4-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 5-Step Order of Operations with Positive & Negative Decimals (Comma Decimal) 6-Step Order of Operations with Positive & Negative Decimals (Comma Decimal)
• Order of Operations With Fractions and Decimals Mixed Order of Operations with Fractions & Decimals Mixed Order of Operations with Fractions & Decimals Mixed with some Negative Values

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The Order of Operations: Examples

PEMDAS Worked Examples Multiplication by Juxtaposition Fractions & Grouping

Most of the issues with simplifying using the order of operations stem from nested parentheses, exponents, and "minus" signs. So, in the examples that follow, I'll be demonstrating how to work with these sorts of expressions.

Content Continues Below

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The Order of Operations

(Links are provided for additional review of working with negatives, grouping symbols, and powers.)

• Simplify 4 − 3[4 −2(6 − 3)] ÷ 2

I will simplify from the inside out. First, I'll simplify inside the parentheses, and then inside the square brackets, being careful to remember that the "minus" sign on the 3 in front of the brackets goes with the 3 . Only once the grouping parts are done will I do the division, followed by adding in the 4 .

4 − 3[4 −2(6 − 3)] ÷ 2

4 − 3[4 − 2(3)] ÷ 2

4 − 3[4 − 6] ÷ 2

4 − 3[−2] ÷ 2

4 + 6 ÷ 2

Remember that, in leiu of grouping symbols telling you otherwise, the division comes before the addition, which is why this expression simplified, in the end, down to " 4 + 3 ", and not " 10 ÷ 2 ".

(If you're not feeling comfortable with all of those "minus" signs, review Negatives .)

• Simplify 16 − 3(8 − 3) 2 ÷ 5

I must remember to simplify inside the parentheses before I square, because (8 − 3) 2 is not the same as 8 2  − 3 2 .

16 − 3(8 − 3) 2 ÷ 5

16 − 3(5) 2 ÷ 5

16 − 3(25) ÷ 5

16 − 75 ÷ 5

16 − 15

If you have learned about variables and combining "like" terms , you may also see exercises such as this:

Simplify 14 x + 5[6 − (2 x + 3)] .

If I have trouble taking a subtraction through a parentheses, I can turn it into multiplying a negative 1 through the parentheses (note the highlighted red " 1 " below):

14 x + 5[6 − (2 x + 3)]

14 x + 5[6 − 1 (2 x + 3)]

14 x + 5[6 − ( 1 )(2 x ) − ( 1 )(+3))]

14 x + 5[6 − 2 x − 3]

14 x + 5[6 − 3 − 2 x ]

14 x + 5[3 − 2 x ]

14 x + 5(3) + 5(−2 x )

14 x + 15 − 10 x

14 x − 10 x + 15

It isn't required that you rearrange the terms to group "like" terms together, but it's generally a good idea to do so, certainly when you're just starting out. But, by grouping appropriately, it's a lot harder to lose terms when you're adding things up.

• Simplify −{2 x − [3 − (4 − 3 x )] + 6 x }

I need to remember to simplify at each step, combining like terms when and where I can. I'll start by inserting the "understood" 1 's in front of the subtracted grouping symbols:

−{2 x − [3 − (4 − 3 x )] + 6 x }

−1{2 x − 1[3 − 1(4 − 3 x )] + 6 x }

−1{2 x − 1[3 − 1(4) − 1(−3 x )] + 6 x }

−1{2 x − 1[3 − 4 + 3 x ] + 6 x }

−1{2 x − 1[−1 + 3 x ] + 6 x }

−1{2 x − 1(−1) −1(+3 x ) + 6 x }

−1{2 x + 1 − 3 x + 6 x }

−1{2 x + 6 x − 3 x + 1}

−1{5 x + 1}

−1(5 x ) − 1(+1)

−5 x − 1

(For more examples of this sort, review Simplifying with Parentheses .)

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Expressions containing fractional forms can cause confusion, too. But, as long as you work the numerator (that is, the top) and the denominator (that is, the bottom) separately, until they're completely simplified first, and only then combine (or reduce), if possible, then you should be fine.

If a fractional form is added to, or subtracted from, another term, fractional or otherwise, make sure you've completely simplified and reduced the fractional form before you try to do the addition or subtraction.

Before I can add the two fractional terms, I first have to simplify each term.

[45]/[8(5 − 4) − 3] + [3(2) 2 ]/[5 − 3]

[45]/[8(1) − 3] + [3(4)]/[2]

[45]/[8 − 3] + [12]/[2]

[45]/[5] + 6

As it happens, each of the fractions above simplified to whole numbers, so I didn't have to muck about with common denominators. You will not usually be so lucky.

To do this simplification, I have to work the top and bottom separately, until I get a fraction that I can (possibly) reduce.

[(3 − 2) + (1 + 2) 2 ]/[5 + (4 − 1)]

[(1) + (3) 2 ]/[5 + (3)]

[1 + 9]/[8]

(For examples with loads of exponents, review Simplifying with Exponents .)

You can use the Mathway widget below to practice simplifying using the order of operations. Try the entered exercise, or type in your own exercise. Then click the button and select "Simplify" or "Evaluate" from the pop-up box to compare your answer to Mathway's. (Or skip the widget and continue to the next page.)

Please accept "preferences" cookies in order to enable this widget.

(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)

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Order of operations

The order of operations is a specific order or a set of rules, agreed upon by mathematicians, one must follow when performing arithmetic operations to simplify expressions.

Order of operations rules

Here is the order for doing operations that you need to follow in the order given below to avoid having different answers when simplifying expressions. If grouping symbols are used such as parentheses, braces, or curly brackets, perform the operations inside the grouping symbols first. Then, proceed with exponents, and so forth...

1. Simplify any expression within parentheses, brackets or grouping symbols: (  ) [  ] {  }

2. Simplify powers or expressions involving exponents: 4 2 , 2 5 , or 5 3

3. Multiply and divide in order from left to right: × and ÷

4. Add and subtract in order from left to right: + and -

Order of operations problems

Study the example in the figure below carefully so that you understand how to use the order of operations!

More examples showing how to use the order of operations

Example #1: 4 2 - 6 × 2 ÷ 4 × 3 + 5

Do exponent:

16 - 6 × 2 ÷ 4 × 3 + 5

Multiply and divide from left to right

16 - 12 ÷ 4 × 3 + 5 16 - 3 × 3 + 5

16 - 9 + 5 Add and subtract from left to right 16 - 9 + 5

Example #2: (2 + 5 2 ) + 4 × 3 - 10

Do parenthesis: (2 + 25) + 4 × 3 - 10

27 + 4 × 3 - 10

Do multiplication

27 + 12 - 10

Example #3:

10 - 14 ÷ 2 = 10 - 7 = 3 (Division comes before subtraction)

Remember that if you see multiplication and division at the same time, perform the operation from left to right.

Example #4:

4 + 5 ÷ 5 × 6 = 4 + 1 × 6 = 4 + 6 =10

How to remember the order of operations

The following acronyms can make it easier for you to remember the order of operations.

• PEMDAS (used mostly in the United States of America and also in France)
• BODMAS (used mostly in UK, Australia, and India)
• BEDMAS (used in Canada and New Zealand)

The following mnemonic may help you remember the PEMDAS rule: PEMDAS ( Please Excuse My Dear Aunt Sally )

• The P stands for parentheses
• The E stands for exponents
• The M stands for multiply
• The D stands for division
• The A stands for add
• The S stands for subtraction

Even though M comes before D in PEMDAS, the two operations have the same precedence. Same precedence means that multiplication is not more important than division. By the same token, even though A comes before S, the two operations have the same precedence. Addition is not more important than subtraction.  

A much better way, in my opinion, to write PEMDAS is P - E - MD - AS.

In P - E - MD - AS, operations with the same precedence have no hyphens between them.

For example, since addition and subtraction have the same precedence, there is no need to put a hyphen between them. 

However, P and E have a hyphen between them because P has a higher precedence than E.

All the four letters in MDAS, DMAS, DMAS, and DMAS refer to multiplication, division, addition, and subtraction. 

• In BODMAS rule, the B stands for bracket and the O stands for order. Order can be powers or roots.
• In BEDMAS rule, the B stands for bracket and the E stands for exponents.
• In BIDMAS rule, the B stands for bracket and the I stands for indices. Indices are powers such as 6 2

Keep in mind also that PEMDAS, BODMAS, BEDMAS, and BIDMAS are all correct ways to perform the order operations. None of them is better than the other. These are just names that are used, based on the country, to make it easier to remember the rules.

What about nested parentheses in the order of operations?

Example #5:

Simplify  √4 + 1 + {2 - [ (6 - 2) × 5] + 13}.

Work first with the innermost set of parentheses or (6 - 2) .

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + {2 - [4 × 5] + 13}

Next, work again first with the inner set of parentheses or  [4 × 5] .

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + {2 - 20 + 13}

Stay inside the parentheses until you are done. While working inside the parentheses, notice that you need to add and subtract in order from left to right.

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + {-18 + 13}

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = √4 + 1 + -5

According to BODMAS rule, you need to do root first.

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = 2 + 1 + -5

Add and subtract again in order from left to right

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = 3 + -5

√4 + 1 + {2 + [(6 - 2) × 5] + 13} = -2

The final answer is -2

A real-life example of PEMDAS

Order of operations quiz. see how you understand this lesson..

Order of operations calculator

Order of operations word problems

Why multiply before adding

Order of operations worksheets

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Project-Based Learning in Fostering Creative Thinking and Mathematical Problem-Solving Skills: Evidence from Primary Education in Indonesia

The interdependence between the Project-Based Learning (PjBL) Model and the growth and enhancement of Creative Thinking and Mathematical Problem Solving Skills in Elementary Schools is unquestionable nowadays. Prior studies have yet to discover concrete evidence regarding the interdependence being discussed. This study highlighted cognitive abilities related to creative thinking and mathematics problem-solving by implementing the Project-Based Learning Model. This research was a quasi-experiment with a pretest-posttest control group design involving 43 students in the sixth grade of two elementary schools; data was collected through test and classroom observation, and then the data was analyzed using Multivariate Analysis of Variance (MANOVA). Conversely, students exposed to project-based learning models exhibit higher skill levels in creative thinking and problem-solving than those instructed using conventional learning models. The project-based learning model significantly impacted elementary school children’s creative thinking and mathematics problem-solving skills. These findings suggest that the Project-Based Learning Model is acceptable for instructors seeking to foster creativity in teaching mathematics at the primary school level in Indonesia or other countries with comparable settings.

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