PEMDAS: The Order of Operations

PEMDAS is a way of remembering the order of operations, which are used to simplify numerical expressions and are even represented in how we approach solving equations. In this lesson, we will review what PEMDAS is and look at several examples of how it works.

Table of Contents

  1. Why do we need PEMDAS?
  2. What does PEMDAS stand for?
  3. Examples of using PEMDAS and the order of operations
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Why do we need PEMDAS?

Consider the calculation \(1+3\times 5\). Which of these is correct?

\(1+3 \times 5 = 4 \times 5 = 20\)

or

\(1 + 3\times 5 = 1 + 15 = 16\)

If you think about it, both seem like reasonable ways to approach the calculation. The problem is, the two approaches give different answers! To avoid confusion, there is a standard agreed upon order in which to perform the operations (where an operation is something like addition or multiplication) known as the order of operations.

By the way, the correct calculation is: \(1 + 3\times 5 = 1 + 15 = 16\)

What is PEMDAS?

The easiest way to remember the order of operations is with the shorthand “PEMDAS”. You can even add in a way to remember this shorthand: “Please Excuse My Dear Aunt Sally”. This stands for:

Parentheses: Perform all operations that are inside parentheses. If there are many operations, you must follow the order of operations inside the parentheses as well.

Exponents: Calculate any exponents you see.

Multiplication and Division: Perform any multiplication or division you see, generally from left to right.

Addition and Subtraction Perform any addition and subtraction you see, from left to right.

Examples applying PEMDAS / the order of operations

Let’s try a couple of examples to see how to apply these rules.

For the following examples, you must remember that an exponent represents multiplication. So, \(3^2 = 9 \times 9\) and \(2^4 = 2 \times 2 \times 2 \times 2\). If you need to review exponents, you can do that here: https://www.mathbootcamps.com/understanding-exponents/

Example

Simplify: \(1 + 3 \times 4^2\)

Solution

Thinking about PEMDAS, we should first start with parentheses. But, since there are no parentheses, we will move on to “E”, which tells us to calculate any exponents. Since \(4^2 = 16\), we can write:

\(1 + 3 \times 4^2 = 1 + 3 \times 16\)

Next, we are to perform any multiplication or division (MD). You can verify with your calculator that \( 3 \times 16 = 48\). Now we have:

\(1 + 3 \times 16 = 1 + 48\)

Finally, we will add/subtract (AS) to get our final answer:

\(1 + 48 = \boxed{49}\)

As you can see, we just need to walk though each part of PEMDAS and do each calculation carefully. Let’s try another example applying the same rules, but just take a look at the math itself.

Example

Simplify: \(3(4-3 \times 10)+2^2\)

Solution

We will follow the same steps as used above. Remember, anything inside of parentheses must be simplified first AND you must follow order of operations within the parentheses.

\(\begin{align} 3(4-3 \times 10)+2^2 &=3(4-30)+2^2\\ &=3(-26)+2^2\\ &=3(-26)+4\\ &= -78+47\\ &= \boxed{-74}\end{align}\)

As you can see, we worked inside of the parentheses first and then followed the order of operations outside of the parentheses once we got down to one number.

Example

Simplify: \(2+3[-2^2+4(2+1)^3]\)

Solution

In this example, there are multiple sets of parentheses. Even though it looks complicated, we still apply the same set of rules. Just start with the innermost parentheses are work your way out.

\(\begin{align} 2+3[-2^2+4(2+1)^3] &= 2+3[-2^2+4(3)^3]\\ &=2+3[-4+4(27)]\\ &=2+3[-4+108]\\ &=2+3[104]\\ &=2+312\\ & = \boxed{314}\end{align}\)

These examples have hopefully shown you how by just applying the order of operations carefully and methodically, you can simplify even the most complicated expressions. Now, we will look at one last example which seems like it should be simple, but shows the importance of parentheses when working with numbers.

Example

Simplify: \(-3^2\)

Solution

This may seem confusing since there is only one operation (the exponent), however there are really two operations here. You can actually write \(-3^2\) as \(-1\times3^2\). Now by the order of operations:

\(-1\times3^2 = -1\times9 = -9\)

In other words \(-3^2 = -9\).

You may be thinking “BUT WAIT I know that anytime you square a negative number you get a positive number!!” . YES! That is true, but here we were not squaring a negative number, we were squaring 3. If we wanted to square –3, we would write \((-3)^2 = 9\). See the difference? If there are parentheses, it means you are squaring the entire term, otherwise the negative “rides along” as part of a multiplication.

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Conclusion

The order of operations applies not just to numerical examples, but also to any mathematical problem including those seen later in algebra and calculus. Make sure you take the time to work with these rules so that they are second nature and don’t slow you down in advanced courses! You should also take special care when using graphing calculators and apply parentheses correctly since they will always use the order of operations when making a calculation.

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