The Order of Operations is very important when simplifying expressions and equations. The Order of Operations is a standard that defines the order
in which you should simplify different operations such as addition, subtraction, multiplication and division.
This standard is critical to simplifying and solving different algebra problems. Without it, two different people may interpret
an equation or expression in different ways and come up with different answers. The Order of Operations is shown below.
Before we begin simplifying problems using the Order of Operations, let’s examine how failure to use the Order of Operations
can result in a wrong answer to a problem.
Without the Order of Operations one might decide to simplify the problem working left to right. He or she would add two and
five to get seven, then multiply seven by x to get a final answer of 7x. Another person might decide to make the problem
a little easier by multiplying first. He or she would have first multiplied 5 by x to get 5x and then found that you can’t
add 2 and 5x so his or her final answer would be 2 + 5x. Without a standard like the Order of Operations, a problem can be
interpreted many different ways.
Below is the first expression that we will be simplifying:
As we walk through the steps to simplifying this expression, use the Order of Operations reference in the right column of
this page. The first step in the Order of Operations is to simplify parentheses and brackets from the inside out. You must
remember to use the Order of Operations when simplifying the inside of the parentheses, but we don’t need to worry about that
in this problem because there is only one operation inside the parentheses 3 – 1. In this case all that has to be done is
subtraction of 1 and 3. The answer is shown below.
The next step in the Order of Operations is to simplify exponents. 3 2 becomes 9. The result is shown below.
The next step in the Order of Operations is to simplify multiplication and division in the order that they appear. There
is no division, only multiplication. Multiply (2) and 9:
The final step is to simplify addition and subtraction (combine like terms).
Below is the next expression that we will be simplifying:
Again, the first step in the Order of Operations is to simplify parentehsis and brackets from the inside out. The polynomial
x + 1 is in the innermost set of parentheses, but nothing inside of it can be simplified. Now we can simplify the second
innermost grouping symbol, the bracket using the Order of Operations.
First simplify multiplication and division in the order they appear. There is no division, all we need to do is distribute
8 into (x + 1).
Now remove the parentheses symbols and simplify addition and subtraction in the order that they appear (combine like terms).
Now, start using the Order of Operations to simplify the polynomial inside of the second set of parentheses. There are no
exponents on the inside so you can skip to simplifying multiplication and division in the order the appear. The division
comes first in this expression, so divide 3 by 2 first.
Multiplication comes second in this problem, so now you can multiply three halves by 2.
Now that the inside of each set of parentheses and brackets are simplified, the problem can be worked on as a whole instead
of in little groups. Now in step two of the Order of Operations, simplify the only exponent in the expression, (3) 2
Continuing with the Order of Operations, multiply [8x + 10] and (9).
There are no terms which can be combined, this problem is complete.
Evaluate the following
First, evaluate whatever is in the parentheses:
Next, evaluate the exponent:
Evaluate any multiplication and division from left to right:
Evaluate any addition and subtraction from left to right or which ever way makes
it easier for you: