The order of operations is one of the most instrumental and basic principles of arithmetic. It refers to the order in which you must perform the various operations in a given mathematical expression. If operations in an expression could be performed in any random order, a single expression would take on a vast array of values. For example:

One student might perform the operations from left to right:

Another student might choose to add before executing the multiplication or division:

As you can see, depending on the order in which we perform the required operations, there are a number of possible evaluations of this expression. In order to ensure that all expressions have a single correct value, we have PEMDAS—an acronym for determining the correct order of operations in any expression. PEMDAS stands for:

Let’s work through a few examples to see how order of operations and PEMDAS work. First, we should find out the proper way to evaluate the expression . Since nothing is enclosed in parentheses, the first operation we carry out is exponentiation:

Next, we do all the necessary multiplication and division:

Lastly, we perform the required addition and subtraction. Our final answer is:

Here’s another example, which is a bit trickier. Try it on your own, and then compare your results to the explanation that follows:

First, resolve the operations under the square root, which is symbolized by and is also called a radical.

But wait, you may be thinking to yourself, I thought we were supposed to do everything within a parentheses before performing exponentiation. Expressions under a radical are special exceptions because they are really an expression within parentheses that has been raised to a fractional power. In terms of math, . The radical effectively acts as a large set of parentheses, so the rules of PEMDAS still apply.

To work out this expression, first execute the operations within the innermost set of parentheses:

Next, perform the required exponentiation:

Then, multiply:

Finally, add:

Now that the operations under the radical have been resolved, we can take the square root.

One additional note is important for the division step in the order of operations. When the division symbolis replaced by a fraction bar (i.e., the expression includes a fraction), you must evaluate the numerator and the denominator separately before you divide the numerator by the denominator. The fraction bar is the equivalent of placing a set of parentheses around the whole numerator and another for the whole denominator.

There are two ways to deal with the order of operations while using a calculator:

If you want to type full expressions into your graphing calculator, you must be familiar with how your calculator works. You can’t enter fractions and exponents into your calculator the way they appear on paper. Instead, you have to be sure to recognize and preserve the order of operations. Practice with the following expression:

If you enter this into a graphing calculator, it should look like this: