+
2 Tutorials that teach The Order of Operations
Take your pick:
The Order of Operations

The Order of Operations

Rating:
Rating
(0)
Description:

In this lesson, students will learn about the Order of Operations

(more)
See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to over 2,000 colleges and universities.*

Begin Free Trial
No credit card required

25 Sophia partners guarantee credit transfer.

221 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 20 of Sophia’s online courses. More than 2,000 colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial
This tutorial covers the order of operations, through the definition and exploration of:
  1. Importance of the Order of Operations
  2. PEMDAS


1. Importance of the Order of Operations

In math, an operation is a way to combine numbers, as in addition or subtraction. 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.

PEMDAS
An acronym used to remember the order of operations: parentheses, exponents, multiplication, division, addition, subtraction
You can use the order of operations to simplify an expression. Suppose you want to simplify the equation:
10 plus 4 divided by 2 minus 1
Using the order of operations, you start with division, dividing 4 by 2, which equals 2. Now your expression is:
10 plus 2 minus 1
You then move on to addition and subtraction. You have 10 plus 2, which is 12, and then 12 minus 1, which is 11.
12 minus 1 equals 11
Conversely, here is what happens if you simplify in the “wrong” order, moving simply from left to right and not using the order of operations. You 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.
table attributes columnalign left end attributes row cell 10 plus 4 divided by 2 minus 1 end cell row cell 14 divided by 2 minus 1 end cell row cell 7 minus 1 equals 6 end cell end table

Therefore, you can see that without having a standard order of operations, you can potentially arrive at two different answers.

The order of operations is used when performing all mathematical calculations, especially when solving equations and evaluating functions.


2. PEMDAS

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

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

There are several important things to remember when using PEMDAS:

  • 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.
Suppose you want to simplify the following expression:
8 minus left parenthesis 7 minus 5 right parenthesis squared plus 3 left parenthesis 2 right parenthesis

You start with your parentheses. In them, you have 7 minus 5, which is 2. Now your expression is:
8 minus 2 squared plus 3 left parenthesis 2 right parenthesis
Note, there is another set of parentheses at the end of the expression around the 2, but in this case, the parentheses are informing you that you multiply the 3 by the 2, an operation that will come later in the process.
The next operation is exponent, so calculate 2 squared, which is 4. Now your expression becomes the following:
8 minus 4 plus 3 left parenthesis 2 right parenthesis
You now move onto multiplication, multiplying 3 times 2, which is 6.
8 minus 4 plus 6
Finally, you have addition and subtraction, which you perform from left to right in the order it appears. First, calculate 8 minus 4, which is 4, and finally, add 6, which equals 10.
4 plus 6 equals 10

The second example involves simplifying an expression containing negative numbers and exponents.

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:
left parenthesis negative 3 right parenthesis squared equals 9 space space space space space minus 3 squared equals negative 9 space space space space
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.
left parenthesis negative 3 right parenthesis squared equals negative 3 cross times negative 3 equals 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.
negative 3 squared equals negative 1 cross times 3 squared equals negative 1 cross times 9 equals negative 9

Now that you know how to avoid this common mistake, try using this knowledge when solving the equation in the second example.

Suppose you want to simplify the following expression:

negative 4 squared plus 12 divided by 2 left parenthesis 3 right parenthesis
You start with your exponent, 4 squared. Remember, the negative in front of the 4 is like a negative 1 being multiplied, so you don’t include it in your exponent operation. Instead, you have 4 squared, which equals 16. Therefore, your expression becomes:
negative 16 plus 12 divided by 2 left parenthesis 3 right parenthesis
You now move on to multiplication and division, which you perform from left to right. First, divide 12 by 2, which equals 6, and then multiply 6 times 3, which equals 18. Finally, you add negative 16 and 18 for a final answer of 2.
table attributes columnalign left end attributes row cell negative 16 plus 6 left parenthesis 3 right parenthesis end cell row cell negative 16 plus 18 equals 2 end cell end table
Today you learned about the importance of the order of operations, which lets us simplify expressions and equations to find the correct answer. You also learned the acronym PEMDAS to remember the order of operations, noting that multiplication and division, as well as addition and subtraction, are performed together from left to right in the order that they appear. Finally, you learned that when raising a negative number to an exponent, parentheses must be used around the negative sign as well.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • PEMDAS

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