Order of Operations
The order of operations 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.
One student might perform the operations from left to
Another student chooses to add before executing the multiplication
As you can see, there are a great many possible evaluations
of this expression depending on the order in which we perform the
required operations. That’s why we have PEMDAS: a catchy 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: before you do any other operation,
raise all the 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
1/3 power, respectively.
- Multiplication and Division:
perform multiplication and division.
- Addition and Subtraction: perform
addition and subtraction.
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 3
4. 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
First, resolve the operations under the square root, which
is symbolized by
and is also called
But wait, you may be thinking to yourself, I thought we
were supposed to do everything within 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:
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 symbol
replaced by a fraction bar (i.e., the expression includes a fraction),
you must evaluate the numerator
and the denominator
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. For example, in the fraction
you must work out the numerator and denominator before
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.
you have a graphing calculator, you can type the whole expression
into your calculator. This method will be faster, but can cause
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. So acquaint yourself by practicing
with the following expression:
When you enter this into your calculator, it should look