What are the Order of Operations?

Order of Operations

Please Excuse My Dear Aunt Sally.

Now, I know what you’re thinking, “What does that phrase actually mean?” Quite a bit, actually, because that saying provides the key to remembering an important math concept: the order of operations.

The order of operations is one of the more critical mathematical concepts you’ll learn because it dictates how we calculate problems. It gives us a template so that everyone solves math problems the same way.

Let’s start off with a simple question. What is an operation?

An operation is a mathematical action. Addition, subtraction, multiplication, division, and calculating the root are all examples of a mathematical operation. Let’s take a look at this problem:

\(7\times 4-6=?\)
.

It looks easy, right? Well, it wouldn’t be so easy if we didn’t understand the order in which the math operation occurs. If we didn’t have rules to determine what calculations we should make first, we’d come up with different answers.

Should you start by subtracting 4 minus 6 and then multiplying by 7?

No. The order of operations tells us how to solve a math problem. And this brings us back to Aunt Sally.

Operations have a specific order, and this is what “Please Excuse My Dear Aunt Sally” helps us to understand. It’s an acronym: PEMDAS that tells us in which order we should solve a mathematical problem.

So first is “Please,” which stands for “Parentheses,” so we solve everything inside of the parentheses first.

Then, “E, Excuse,” which is for “Exponents.” We solve that after we solve everything in parentheses.

Multiplication, which is the “My,” and this happens from left to right.

And then division, which is the “Dear,” which also happens left to right.

And then we have addition and subtraction which also happens from left to right, and this is “Aunt” and “Sally.”

Okay, so now that we know the order of operations, let’s apply it to our problem that we have here and solve.

\(7\times 4-6=?\)
.

So if, let’s kinda go down our list. We don’t have parentheses and we don’t have exponents, but we do have multiplication so we do that before we do any addition or subtraction. So let’s go ahead and multiply \(7\times 4\). That gives us 28.

\(28-6\)
.

And now we’re subtracting 6, which gives us 22.

\(28-6=22\)
.

Now, let’s look at another problem.

\(7+7\times 3\)
.

Without the operations, you could calculate this problem as \(7+7=14\times 3=42\).

And this would be wrong!

Remember, you multiply before you add. Therefore, the equation should look like this:

\(7+(7\times 3)\)
.
\(=7+21\)
.
\(=28\)
.

So when we do problems like this, we can use parentheses to group together our numbers that are going to take place first. So in this case, it’s \(7\times 3\), and when we do that we get 21 and we have plus 7 left over. When we add those together, we get 28, and that’s our answer!

Let’s look at some more complex problems.

\(6\times 3^{2}\)
.

The order of operations dictates how to solve this problem. Remember, you multiply exponents first. Here’s the wrong way to solve the problem:

\(6\times 3^{2}\)
.
\(=(6\times 3)^{2}\)
.
\(=18^{2}\)
.
\(=324\)
.

Why is that wrong? Because you violated the order of operations. You do not multiply first! You perform an operation on the exponent first. This is how it should be done:

\(6\times (3^{2})=\)
.
\(6\times 9\)
.
\(=54\)
.

See? Solving the equation in the right order provides the correct answer.

Let’s try out one more problem. This one is a little bit more challenging, but it perfectly illustrates the order of operations.

\(5\times 10-(8\times 6-15)+4\times 20\div 4\)
.

Remember the order. What do we do first? The numbers inside the parentheses. So \(8\times 6=48\), then we subtract 15 and that gives us 33. Here’s how the problem looks now:

\(5\times 10-33+4\times 20\div 4\)
.

So our next step is multiplication and division, so let’s perform all our multiplication and division problems and then see what we have left.

\(50-33+80\div 4\)
.
\(50-33+20\)
.

Now we finish with addition and subtraction, so here’s what we have:

\(50-33+20\)
.
\(=50-13\)
.
\(=37\)
.

And our answer is 37!

There is an exception. If an equation only has one expression, you don’t have to follow the order of operations.

Here are some examples of single expressions.

\(10+10\): Well, there are no other operations, so you just know to go ahead and add them together and you get 20. Same thing with subtraction, multiplication, and division. All those are single expressions.

Alright guys, that’s our video on the order of operations. I hope that this was helpful!

See you guys next time!

Frequently Asked Questions

Q

What is the order of operations in math?

A

The order of operations is the order you use to work out math expressions: parentheses, exponents, multiplication, division, addition, subtraction. All expressions should be simplified in this order. The only exception is that multiplication and division can be worked at the same time, you are allowed to divide before you multiply, and the same goes for addition and subtraction. However, multiplication and division MUST come before addition and subtraction. The acronym PEMDAS is often used to remember this order.

Ex. Use the order of operations to simplify the expression \(3×4^2+8-(11+4)^2÷3\).

Parentheses: \(3×4^2+8-(15)^2÷3\)

Exponents: \(3×16+8-225÷3\)

Multiplication/Division: \(48+8-75\)

Addition/Subtraction: \(-19\)

Q

Do you use the order of operations when there are no parentheses?

A

Yes, always use the order of operations to simplify expressions. If there are no parentheses, then skip that step and move on to the next one. The same applies for any other missing operation.

Ex.Use the order of operations to simplify the expression \(6^2-4+2\).

Parentheses: There are none, so skip this step.

Exponents: \(36-4+2\)

Multiplication/Division: There isn’t any, so skip this step.

Addition/Subtraction: \(34\)

Q

Do calculators do order of operations?

A

No, most calculators do not follow the order of operations, so be very careful how you plug numbers in! Make sure you follow the order of operations, even if that means plugging in numbers in a different order from how they look on your page.

Q

Which math operation comes first?

A

Parentheses are the first operation to solve in an equation. If there are no parentheses, then move through the order of operations (PEMDAS) until you find an operation you do have and start there.

Q

What are the basic operations?

A

The four basic operations are: addition (+), subtraction (-), multiplication (×), and division (÷).

Practice Questions

Question #1:

 
\(7\times9+3-6\div2+2^2-11\)

47

56

23

13

Answer:

The correct answer is 56. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.
\(7×9+3-6÷2+2^2-11\)
There are no parentheses in this problem, so start with exponents.
\(7×9+3-6÷2+4-11\)
Then, multiply and divide from left to right.
\(63+3-6÷2+4-11\)
\(63+3-3+4-11\)
Finally, add and subtract from left to right.
\(66-3+4-11\)
\(63+4-11\)
\(67-11\)
\(56\)

Question #2:

 
\(19+7(26-48÷2)^3+3×6\)

93

1,902

173

251

Answer:

The correct answer is 93. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.
\(19+7(26-48÷2)^3+3×6\)
First, start with parentheses. The order of operations must be followed even inside parentheses, so be sure to divide before you subtract.
\(19+7(26-24)^3+3×6\)
\(19+7(2)^3+3×6\)
Next comes exponents.
\(19+7(8)+3×6\)
Then, multiply from left to right.
\(19+56+3×6\)
\(19+56+18\)
Finally, add from left to right.
\(75+18\)
\(93\)

Question #3:

 
\(11+3-7×2+1×4÷2\)

26

19

4

2

Answer:

The correct answer is 2. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.
\(11+3-7×2+1×4÷2\)
There are no parentheses or exponents, so start with multiplication and division from left to right.
\(11+3-14+1×4÷2\)
\(11+3-14+4÷2\)
\(11+3-14+2\)
Finally, add and subtract from left to right.
\(14-14+2\)
\(0+2\)
\(2\)

Question #4:

 
\(3(11+2)^2-18÷6\)

504

201

81

127

Answer:

The correct answer is 504. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.
\(3(11+2)^2-18÷6\)
First, simplify what is in parentheses.
\(3(13)^2-18÷6\)
Then, do any exponents.
\(3(169)-18÷6\)
Next, multiply and divide from left to right.
\(507-18÷6\)
\(507-3\)
Finally, subtract.
\(504\)

Question #5:

 
\((16-24)^2+3×11-1\)

736

451

96

72

Answer:

The correct answer is 96. The order of operations can be remembered by the acronym PEMDAS, which stands for: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right.
\((16-24)^2+3×11-1\)
First, simplify the parentheses.
\((-8)^2+3×11-1\)
Then, do exponents.
\(64+3×11-1\)
Next, multiply.
\(64+33-1\)
Finally, add and subtract from left to right.
\(97-1\)
\(96\)

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by Mometrix Test Preparation | Last Updated: June 7, 2021