Order of Operations

1. Importance of the Order of Operations

Have you ever seen something like this in your social media feed with the challenge to solve it?

What is the answer? Is it 11? 6? 14? You see multiple different answers given and everyone insists they have the right answer. Why is that?

Well, in math, an operation is a way to combine numbers, as in addition or multiplication. You can think of an operation in math as a calculation between two or more numbers.

There needs to be an agreed upon order for performing operations so that when there are several operations in an expression or an equation, everyone simplifies or solves in the same way to get the correct answer. Therefore, the order of operations is the rule that tells you the order in which to perform those operations.

The correct order of operations is Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction, otherwise referred to by the acronym PEMDAS.

hint

Want an easy way to remember “PEMDAS”? Create a phrase that can help you remember such as: Please Excuse My Dear Aunt Sally. You can use this phrase or come up with one of your own!

term to know

PEMDAS

An acronym used to remember the order of operations: parentheses, exponents, multiplication, division, addition, subtraction.

2. PEMDAS

PEMDAS is the acronym you can use to remember the order of operations. PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

• Parentheses include other grouping symbols, such as brackets or radical signs .
• Multiplication and division are performed together from left to right in the order that they appear.
• Similarly, addition and subtraction are performed together from left to right in the order that they appear.

hint

Think of each part of PEMDAS as a “level.” First, scan the expression for any parentheses. If there are any, complete all of the operations inside before moving on to the next level. At the next level, complete all multiplication and division sections (from order of left to right). Once that is solved, move on to addition and subtraction (from order of left to right).

You can use the order of operations to simplify an expression.

EXAMPLE

Suppose you want to simplify the expression:

	The expression
	Since there are no parentheses, you can move to the next level and start with division; divide 4 by 2.
	Now that the multiplication/division level is complete, only the addition and subtraction operations are remaining. Because these are on the same level, complete the addition and subtraction operations from left to right; first, add 10 and 2.
	Find the solution by subtracting 1 from 12.

If this had been on your social media feed, and one of your friends solved the expression without using the proper order of operations, they may have just tried to solve it from left to right. If they would have done that, they would start with 10 plus 4, which equals 14; 14 divided by 2 is 7, and 7 minus 1 is 6, which is an incorrect answer.





This is why you find so many incorrect answers to these social media challenges. You can see that without having a standard order of operations, you can potentially arrive at many different answers with everyone insisting they are correct.

try it

Suppose you have the following expression: .

How would you simplify this expression?

.	The expression
	Evaluate the equation inside of the parentheses; 7 minus 5 is 2. Why don’t we do the other parentheses with the 3(2)? This is because there are no operations to complete inside or between the numbers—we assume this represents multiplication. We’re not at that level yet, so wait for now.
	Next, we calculate the exponent; is 4.
	Now, we move onto multiplication; 3 times 2 is 6.
	Finally, addition and subtraction from left to right; 8 minus 4 is 4.
	Add 4 and 6 to get a solution of 10

The original expression had two sets of parentheses. The set of parentheses at the end of the expression (around the 2) are informing you that you multiply the 3 by the 2, an operation that will come later in the process.

3. Common Mistakes

Before we move on, it is important to note a common mistake that people make when solving or simplifying expressions containing negative numbers and exponents. Consider the two similar, but different, equations or statements below:

In the first equation, you have negative 3 in parentheses squared, which equals a positive 9. Negative 3 squared means negative 3 times negative 3, which is a positive 9.

In the second equation, you have a negative 3 squared, which equals negative 9. That’s because the negative here is like a negative 1 being multiplied by the 3 squared. Therefore, the answer becomes negative 9.

try it

Suppose you have the following expression: .

How would you simplify this expression?

.	The expression
	Evaluate the exponent; (the negative in front is like a negative 1 being multiplied, so you do not include it in your exponent operation).
	Next we move onto multiplication and division, moving from left to right; 12 divided by 2 is 6.
	Multiply 6 by 3, which is 18.
	Add negative 16 and 18 to get a solution of 2