Order of Operations Lessons
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.
- Parentheses and Brackets -- Simplify the inside of parentheses and brackets before you deal with the exponent (if any) of the set of parentheses or remove the parentheses.
- Exponents -- Simplify the exponent of a number or of a set of parentheses before you multiply, divide, add, or subtract it.
- Multiplication and Division -- Simplify multiplication and division in the order that they appear from left to right.
- Addition and Subtraction -- Simplify addition and subtraction in the order that they appear from left to right.
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.
Order of Operations Example
Below is the first expression that we will be simplifying:
Order of Operations
- Parentheses and Brackets
from the inside out -
Exponents
of numbers or parentheses -
Multiplication and Division
in the order they appear. -
Addition and Subtraction
in the order they appear.
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).
Order of Operations Example (2)
Below is the next expression that we will be simplifying:
Order of Operations
- Parentheses and Brackets
from the inside out -
Exponents
of numbers or parentheses -
Multiplication and Division
in the order they appear. -
Addition and Subtraction
in the order they appear.
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).
Order of Operations
- Parentheses and Brackets
from the inside out -
Exponents
of numbers or parentheses -
Multiplication and Division
in the order they appear. -
Addition and Subtraction
in the order they appear.
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.
Examples of the Order of Operations
Example 1
Evaluate the following
Step 1
First, evaluate whatever is in the parentheses:
Step 2
Next, evaluate the exponent:
Step 3
Evaluate any multiplication and division from left to right:
Step 4
Evaluate any addition and subtraction from left to right or which ever way makes it easier for you:
note that
is evaluated as
Example 2
Solve for x in the equation below
Step 1
As always evaluate the expression within the parentheses first, since there is more than one operator in the parentheses, apply PEMDAS to the expression
Step 2
Divide both sides by 21
Step 3
Example 3
Evaluate the following
Step 1
note that in the above expression, one can choose to divide first since that makes the computation easier
Step 2
Example 4
Solve for x in the equation below
Step 1
Step 2
Step 3
Step 3
Quiz on the Order of Operations
Step 1: Evaluate the inner parentheses first
Step 2: Evaluate the multiplication within the parentheses
Step 3: Evaluate the subtraction within the parentheses
Step 4: Finally evaluate the last multiplication
Order of Operations Resources
Practice Problems / Worksheet Order of Operations Calculator
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