Order of operations

The order of operations is a very simple, but a very important rule. So important, in fact, that you will have to obide by it in every math problem you encounter – ever! It concernes the order in which these operations need to be performed. The order of operations lists mathematical operations in order in which they should be performed in a mathematical expression, from highest to lowest priority. Here is that list:

order of operations

The first letters of these operations (P-E-M-A) are often used as a mnemonic device to help you remember the order in which they should be performed. As you can see, operations inside parentheses have the highest priority, then exponents, then multiplication and, with the lowest priority, addition. To explain how this translates to problem solving, we will use this example:

(8 – 10)2 – (-9) * (-3)

This is a fairly complex example, but it contains all the elements needed to properly explain the order of operations . The first step we have to do is to get rid of the parentheses where it is possible. That can be done by solving the expressions inside the parentheses.

(-2)2 – (-9) * (-3)

Notice that we left -9 and -3 inside their parentheses. That is because there are no operations to perform there and we will get rid of them a bit later. We will leave them be to avoid clutter. Now we need to perform the exponentation, as it is second on the list of priorities. Since (-2)2 equals 4, we get:

4 – (-9) * (-3)

The next operation on the order of operations is multiplication, so let us do it. The result of multiplying (-9) and (-3) is 27, so we are left with:

4 – 27

The only operation left to perform now is subtraction, which has the lowest priority. The end result is:

-23

It is important to adhere to this order as it affects the end result. If we performed the operations in a different order, we could have gotten the wrong result.

If you wish to practice the order of operations, you can use the free worksheets below.

Order of operations exams for teachers

Exam Name File Size Downloads Upload date
Non positive numeric expression
Order of operations – non positive integers 725.4 kB 4358 October 13, 2012
Order of operations – non positive decimals 602.6 kB 1747 October 14, 2012
Order of operations – non positive fractions 901.4 kB 1741 October 13, 2012
Order of operations – non positive mixed numbers 933.8 kB 1497 October 13, 2012
Order of operations – non positive improper fractions 857.5 kB 1393 October 13, 2012
Positive numeric expression
Order of operations – positive integers 662.2 kB 4820 October 13, 2012
Order of operations – positive decimal 732.4 kB 1591 October 13, 2012
Order of operations – positive fractions 848.8 kB 2319 October 13, 2012
Order of operations – positive mixed numbers 878.5 kB 1733 October 13, 2012
Order of operations – positive improper fractions 794.9 kB 1406 October 13, 2012
Non positive algebraic expression
Order of operations – non positive integers 897.4 kB 2078 October 13, 2012
Order of operations – non positive decimals 897.4 kB 1226 October 13, 2012
Order of operations – non positive fractions 1.2 MB 1143 October 13, 2012
Order of operations – non positive mixed numbers 1 MB 1246 October 13, 2012
Order of operations – non positive improper fractions 1020.2 kB 1160 October 13, 2012
Positive algebraic expression
Order of operations – positive integers 865.7 kB 2605 October 13, 2012
Order of operations – positive decimals 982.5 kB 1298 October 13, 2012
Order of operations – positive fractions 1.2 MB 1449 October 13, 2012
Order of operations – mixed fractions 1 MB 1425 October 13, 2012
Order of operations – improper fractions 981.2 kB 1330 October 13, 2012

Order of operations worksheets for students

Worksheet Name File Size Downloads Upload date
Order of operations – Positive numeric expressions 781.9 kB 4427 October 14, 2012
Order of operations – Non positive numeric expressions 843.8 kB 2636 October 14, 2012
Order of operations – Positive algebraic expressions 924.8 kB 2321 October 14, 2012
Order of operations – Non positive algebraic expressions 1.4 MB 2118 October 14, 2012

Order of operations knowledge test