


Order of Operations
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:
 Parentheses: first, perform the operations in the innermost parentheses. A set of parentheses supercedes any other operation.
 Exponents: raise any required bases to the prescribed exponent. Exponents include square roots and cube roots, since those two operations are the equivalent of raising a base to the ^{1}⁄_{2} and ^{1}⁄_{3} power, respectively.
 Multiplication and Division: perform multiplication and division.
 Addition and Subtraction: perform these operations last.
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.
Order of Operations and Your Calculator
There are two ways to deal with the order of operations
while using a calculator:
 Work out operations one by one on your calculator while keeping track of the entire equation on paper. This is a slow but accurate process.
 If you have a graphing calculator, you can type the whole expression into your calculator. This method will be faster, but can cause careless errors.
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:
