For teachers: Printable Study Guide and Activity
Simplifying Expressions
Some expressions only contain one operation and are pretty easy to simplify (operations are just things like addition, subtraction, multiplication, etc.). For example, to simplify the expression 2 + 3, you just add 2 and 3 together and get 5 as the answer.
What happens when an expression contains more than one operation? Things get just a little more complicated. Does it make a difference which operation you do first? Let's look at an example with two operations: addition and multiplication.
What happens when an expression contains more than one operation? Things get just a little more complicated. Does it make a difference which operation you do first? Let's look at an example with two operations: addition and multiplication.
Will it make a difference if you add first or multiply first? Let's see.
If you add first, you end up with 25 as the answer. If you multiply first instead, you get 17 for the answer. Which one is correct?
Mathematicians discovered this sort of problem hundreds of years ago. To avoid confusion and multiple answers, they developed what's called the "order of operations." Basically, everyone got together and they all agreed to simplify expressions in the same order. This way, expressions will only have one answer and there won't be any confusion about which one to do first.
Open the next tab to see what the order is.
The Order of Operations (PEMDAS)
Way back when, mathematicians developed a set of rules to follow for simplifying expressions. They decided that they would all follow the same order to make things easier. If you have an expression with more than one operation, this is the order you need to follow:
An easy way to remember the order is to remember the phrase "Please Excuse My Dear Aunt Sally." There are a couple different ways to describe the order of operations. For example, some teachers might say Brackets instead of Parentheses or Indices instead of Exponents. Don't worry, it's all the same order. Let's look at the steps more closely.
Step 1: Parentheses (or other grouping symbols)
If there are any operations inside parentheses, do those first. For example, to simplify 3(6 - 2) you would simplify subtract first since it's inside parentheses and then multiply.
Parentheses are just one way to group things together. You might also see brackets like [ ] or { }. Operations inside any of these grouping symbols should be done first. If there are more than one type of parentheses or brackets, simplify what's in the innermost ones first and work your way outside.
Step 2: Exponents (or roots)
After simplifying operations inside parentheses or brackets, simplify any exponents next. At this step, you also simplify any roots that you may have (for example, a square root).
Some teachers, especially in Europe, refer to this step as "Orders." Orders are the same as exponents. You might see the expression BODMAS instead of PEMDAS for the order of operations. This is just because they're calling the steps Brackets and Orders instead of Parentheses and Exponents.
Step 3: Multiplication or Division (from left to right)
Next, simplify any multiplication or division. If the expression has more than one multiplication or division sign, work your way from left to right.
Important Note: Don't assume you multiply first since it's listed first in PEMDAS. Do which ever one comes first from left to right just like how you read a book. In some expressions you'll multiply first and in others you'll need to divide first.
Step 4: Addition or Subtraction (from left to right)
Last, simplify any addition or subtraction. Just like with the previous step, work your way from left to right.
Open the next tab to see some examples.
Order of Operations Examples
This expression has three different operations: subtraction, multiplication, and addition. A common mistake is for students to just work from left to right. Make sure to follow the order of operations! Go step by step. Any parentheses or grouping symbols? No. Any exponents? No. Any multiplication or division? Yes. This means you need to multiply first. Multiply the 3 and 4 together and bring down the rest of the expression. Now you have addition and subtraction. Addition and subtraction are together in the same step. If you have both, you do the one that comes first from left to right. This means you need to do the subtraction and then the addition.
The most common mistake on a problem like the one above is to add and then subtract. Make sure to remember that addition and subtraction go together in the same step. Work your way from left to right if you have both addition and subtraction (or if you have multiplication and division).
Start with Step 1: Any parentheses or grouping symbols? Yes. Start by simplifying what's inside the parentheses. There are two operations inside the parentheses: subtraction and an exponent. Which one comes first in the order of operations? Exponents. This means you need to simplify 3 squared first and then subtract. This leaves you with two operations left: division and multiplication. Which one comes first in the order of operations? Division and multiplication go together in the same step, so you do the one that comes first from left to right. In this problem, we need to divide first and then multiply.
This example has two sets of grouping symbols: parentheses () and brackets []. When an expression has more than one set of grouping symbols, start with the innermost symbols and work out from there. In this case, we need to start with the 6 squared inside the inner parentheses. Once we simplify that, we need to keep simplifying what's inside the brackets next. This gives you 6 + 4[9]. At this point, you're left with two operations: addition and multiplication. Multiplication comes first in the order of operations, so you need to multiply 4 and 9 together first and then add 6 as the last step.
A common mistake in the example above is to multiply the 3 and 6 together and then square it. Make sure to remember that exponents come before multiplication.
Open the next tab to try some problems on your own.
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