PEMDAS: Math Order of Operations Acronym Explained 

Few would argue that math is a complicated science: all these signs, numbers, fractions, and so many other things. Oh, and what if one example has multiple operations (e.g., parenthesis, division, multiplication, and addition)? 

How do you know where to start, and what’s the right order of operations? Don’t panic; there is a  PEMDAS rule for such cases. Aren’t familiar with it yet? Then, you are in the right place because we will tell you how this acronym can make your math life easier.

What Is the PEMDAS Rule?

You’re probably already wondering how a few letters can make solving math problems easier. It’s all about the order of operations; let’s look closer.

What Does PEMDAS Stand For?

First things first, PEMDAS is an acronym that describes the correct order of mathematical operations in complex examples. Thus, every letter means a particular operation. So, PEMDAS stands for:

• P: Parenthesis
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Source: CSUSM

You may ask, ‘What’s so special about it?’ The point is that the proper order of operations determines the correctness of the answer itself. Any other order of addition, subtraction, multiplication, and division is wrong. 

BEDMAS vs. PEMDAS

You may also find another acronym – BEDMAS. If in PEMDAS rules, the first letter stands for parentheses, here it stands for brackets. Hence the meaning of BEDMAS: brackets, exponents, division, multiplication, addition, and subtraction. 

Also note that in this case, division comes first, then multiplication. Another difference is that the acronym PEMDAS is predominantly used in the US. Meanwhile, the term BEDMAS is used in the UK, Canada, and many other countries.

BODMAS vs. PEMDAS

You may also come across another acronym called BODMAS. It’s very similar to BEDMAS. The difference is that the second letter stands for ‘orders.’ What does that mean? These orders can be a square root or an exponent. This variant can also be found in the UK. Well, as you can see, it’s essentially the same acronym.

How to Remember the Meaning of PEMDAS?

It doesn’t sound complicated, does it? Still, how do you memorize this acronym? The simplest option is to correlate each letter with a particular mathematical operation. But this may sound a bit boring. So, there is a more straightforward tip. 

And it goes like this: Please Excuse My Dear Aunt Sally. This phrase will immediately help you understand and remember the essence of the PEMDAS rule. 

If this method of memorization looks strange and ineffective to you, you can invent your own memory tricks. The main thing is that it should do its job.

Check How to Use the PEMDAS Rule With Examples

So, let’s now talk about how this math acronym can help you solve equations and win over math problems. So what is in store for you in this section? Don’t forget that every rule has its exceptions and cases when its use doesn’t make sense.  

When to Use PEMDAS and When Not to?

When to use the PEMDAS rule? It’s quite simple: when there are two or more of the above mathematical operations in one example. Here are step-by-step instructions:

1. Solve the equation in parentheses: round (), square [], or curly {}.
2. Exponential operations should be calculated before the multiplication, division, addition, and subtraction. They are usually expressed in powers or roots.
3. Then, perform multiplication or division. Follow the direction from left to right, whichever comes first in the equation.
4. Make addition or subtraction, whichever comes first, also moving from left to right.

Okay, are there any cases when PEMDAS does not work? Yes, there are a few: 

• Ambiguity. If enough parentheses are missing from an equation. For example, let’s take the expression 6 ÷ 2(1 + 2). Someone may interpret it as (6 ÷ 2) * (1 + 2) = 9, and others may interpret it as 6 ÷ [2(1 + 2)] = 1. This situation is awkward, especially if you face such an example in an exam. 
• Complex functions. PEMDAS can help you with equations that have several different mathematical operations. But if these are trigonometry examples or matrices, you will need to follow some other rules. 
• Special designations. Some mathematical expressions do not fit into the PEMDAS framework. Consider the factorial (!) as an example. 5! is calculated as 5! = 5 × 4 × 3 × 2 × 1 = 120, which does not involve standard arithmetic operations.

PEMDAS Rule Examples to Always Get a Correct Answer

PEMDAS Rule: Example #1

So now, let’s figure out how the PEMDAS rule in math works in practice, helping to simplify the expressions. Suppose you have the following example: 

What is the right order of operations in this case? There are a few things that might be a bit confusing. So, where do you start start?

1. First, you need to solve the brackets (3 x 5) = 15. Now, we have 42 ÷ 15 + 3².
2. Second, try performing exponent (3²) = 9. After this manipulation, the original example looks like 42 ÷ 15 + 9. 
3. Next, perform the division (42 ÷ 15)= 2.8.
4. Finally, adding the two remaining numbers gives the following final answer  2.8 + 9 = 11.8.

As you might guess, the PEMDAS rule helps to solve this task quickly and easily. There is no need to memorize long mathematical formulas; just use the acronym to understand what order of operations to follow. Let’s check some more examples.

PEMDAS Rule: Example #2

Earlier, we said PEMDAS might be ineffective when the equation lacks brackets. But what to do when there are too many of them? 

Let us imagine the following example: 

As you can see, there are as many as three types of parentheses. This can be confusing. But don’t worry, PEMDAS can handle it. In this example, you should follow this order of operations: 

1. So, you should first perform operations in round (), second in curly {}, and finally in square [] brackets.
2. After the first manipulation, our equation will take the following form: [23 + {14 – 21}].
3. Then you work with the curly brackets and get [23 + { -7}].
4. Finally, do a simple calculation in square brackets: 23 – 7.
5. The one correct answer to this example is 16. 

It’s not so scary, is it?

PEMDAS Rule: Example #3

So, if your expression contains square roots, when do you solve them? There is no direct answer to this question in the acronym. Let’s consider the following example to understand the order of operations we need:

1. A square root in an equation should be evaluated during the ‘E’ step, which stands for ‘Exponent’ in PEMDAS. If there are two parts with exponents, they are solved independently. Thus, you first solve the parts: ?(25) = 5 and 3² = 9.
2. Finally, you have two numbers that need to be added together: 5 + 9. So you get the final answer: ?(25) + 3² = 14.

So, now you understand better how the order of operations works in practice.

Common PEMDAS Mistakes: How Not to Get the Incorrect Answer

PEMDAS is a simple rule to solve math problems. Still, you can make mistakes when it comes to choosing a proper order of operations in practice. Here are the most common cases. Let’s find out how to avoid them. 

Source: Pexels

• Wrong order. Sometimes, because of the rush, you can confuse the order of operations. For example, multiplication should be done first and only then addition. If you do it the other way around, you will get the wrong answer. Consider this simple equation: 2 + 3 × 4. If we add the two numbers and then multiply them, we get 20, which is incorrect. The correct answer is 14, which results from multiplication and then addition. As you can see, the PEMDAS rule for order of operations is critical. 
• Confusion of multiplication and division. For example, you may solve the equation 6 ÷ 2 × 3 as (6 ÷ 2) × 3 = 9 instead of the correct 6 ÷ (2 × 3) = 1. Remember to solve multiplication and division from left to right. Thus, in the correct order, 2 × 3 is performed first, then the result is divided by 6.
• Underestimation of parentheses. Let’s check the next example 3 × (2 + 5)². You are wrong if you think the correct answers are this 3 × 2 + 5² = 31 or something like this. The correct one is 3 × (2 + 5)² = 3 × 7² = 147. Why? Because of the order of operations: brackets, exponents, division, or multiplication.

How to Avoid Wrong Answer When Using the PEMDAS Rule?

How can you avoid these issues and always get the right answer when determining the right order of operations? First of all, you should have a good understanding of what the PEMDAS rule is and how it solves math problems. Remember it and practice a bit. If you still need help, consider studying with a tutor. 

You’ll master the theory and work through all the controversial points in more detail. In addition, many online educational services are available today to help you achieve the desired results, which is especially relevant when preparing for a math exam.

Conclusion

PEMDAS, an acronym for math order of operations, is a good tool to help you remember the sequence of basic operations for solving math equations. It will be extremely useful to solve many math problems while preparing for the exam. If you master this rule, you will not be confused by equations with several different operations. In the end, the correct order of operations plays a significant role in arithmetic expressions, and that’s where PEMDAS is your savior.