# Order of Operations

Many people use memory tools like PEMDAS to remember the order of operations in math. However, I think it is easier to learn the logic behind the order of operations and then follow two simple rules:

1. ﻿Grouping symbols trump everything.
2. Do the most difficult operation first.

So, what is an operation? Addition, subtraction, multiplication, and division are the most common operations in math. But there are also other less-used operations like exponents and roots.

How do you determine which is the “most difficult operation”? Let’s take a look at how all the operations are related to each other.

When we first started learning math, we learned about numbers by learning how to count. We also learned how to “un-do” counting by counting backwards.

After we mastered addition and subtraction, we learned multiplication. Multiplication also made life easier because it gave us a short cut for repeated addition. Eventually, we learned how to undo multiplication with division.

Finally, after mastering addition and multiplication, we learned about exponents. Positive exponents gave us a way to write repeated multiplication. You may not have learned about roots yet, but roots are what we use to undo exponents.

## The Pyramid of Operations

There are four layers of mathematical operations (counting, adding, multiplying, and exponents).

Each of the four layers has an inverse operation that undoes the main operation (backwards counting, subtraction, division, roots).

I like to think of these four layers as a pyramid because they build on each other.

Exponents were created by repeated multiplication.

Multiplication was created by repeated addition.

Addition was created by repeated counting.

This layered structure is how we define which operation is the "most difficult" when we are doing order of operations.

Hypothetically, we could create another layer on this pyramid and invent something to represent repeated exponents.

However, mathematicians haven't invented an operation for repeated exponents yet because we just don't use them very often. And when we do use repeated exponents, we usually simplify them using the power rule.

Maybe you'll be the mathematician that convinces the world that repeated exponents should have their own operation ;)

## How to Use Order of Operations in Math

When you are using order of operations, all you need to do is follow these two rules:

1. Grouping symbols trump everything.
2. Do the most difficult operation first.

If there are no grouping symbols in the expression, you should start with the most difficult level of the pyramid (exponents and roots).

After you have simplified all the exponents and roots, move on to the next level of the pyramid (multiplication and division). It does not matter whether you do multiplication or division first.

Once the multiplication and division is done, you can move on to the last level of the pyramid (addition and subtraction). When you have addition and subtraction in the same problem, make sure to work from left to right

## Example: No Grouping Symbols

$$7 \times 5 - 8 \div 2 + 3^{4}$$

1. Are there any grouping symbols? No.

2. What is the most difficult operation? Exponents, so I will simplify my exponent first.

$$7 \times 5 - 8 \div 2 + \color{black}{3^{4}}$$

$$3^{4}=3\times 3\times 3\times 3 =81$$

3. What is the next most difficult operation? Multiplication and division are the same level of difficulty and it doesn't matter which one I do first so I'll start with multiplication.

$$\color{black}{7 \times 5} - 8 \div 2 +81$$

$$7 \times 5 = 35$$

Then I'll do division.

$$35- \color{black}{8\div2}+81$$

$$8 \div 2 =4$$

4. What is the next most difficult operation? Addition and subtraction are the same level of difficulty and I need to calculate them from left to right, so I will start with subtraction.

$$\color{black}{35 - 4} + 81$$

$$35 - 4 =31$$

Then I'll finish the last operation (addition).

$$31 + 81 = 112$$

If there are any grouping symbols like parentheses (), brackets [], or braces {}, remember that grouping symbols trump everything. The operations inside the grouping symbols have to be done first even if they are on a lower level of the math pyramid.

## Example: Grouping Symbols

$$(8-2) \times 3^{2} \div 2 + 1 - 5$$

1. Are there any grouping symbols? Yes, so I will simplify the subtraction first even though it is the easiest operation.

$$\color{black}{(8-2)} \times 3^{2} \div 2 + 1 - 5$$

$$8-2=6$$

2. Now that the grouping symbols are simplified, what is the most difficult operation? Exponents, so I will simplify my exponent next.

$$6\times \color{black}{3^{2}} \div 2 + 1 - 5$$

$$3^{2}=3\times 3=9$$

3. What is the next most difficult operation? Multiplication and division are the same level of difficulty. It doesn't matter which one I do first but I'll start with multiplication.

$$\color{black}{6 \times 9} \div 2 + 1 - 5$$

$$6 \times 9 = 54$$

Then I'll do division.

$$\color{black}{54 \div 2} + 1 - 5$$

$$54 \div 2 =27$$

4. What is the next most difficult operation? Addition and subtraction are the same level of difficulty and I have to calculate them from left to right, so I will start with addition.

$$\color{black}{27 + 1} - 5$$

$$27 + 1 =28$$

Then I'll finish the last operation (subtraction).

$$28 - 5 = 23$$

If you have multiple operations inside the grouping symbols, follow the order of operations within the grouping symbols until everything inside the grouping symbols is simplified, then continue with the normal order of operations outside the grouping symbols.

## Example: Grouping Symbols with Multiple Operations

$$12 \div 4 + 5 - (3^{3} - 8 \times 3)$$

1. Are there any grouping symbols? Yes, and there are multiple operations inside the grouping symbols. So, I will start with the most difficult operation inside the grouping symbols (exponents).

$$12 \div 4 + 5 - (\color{black}{3^{3}} - 8 \times 3)$$

$$3^{3}=3\times 3\times 3=27$$

Then I will do the next most difficult operation inside the grouping symbols (multiplication).

$$12 \div 4 + 5 - (27 - \color{black}{8 \times 3})$$

$$8\times 3=24$$

Lastly, I will simplify the addition so that the grouping symbol part of the expression is completely simplified.

$$12 \div 4 + 5 - \color{black}{(27 -24)}$$

$$27-24=3$$

2. Now that the grouping symbols are simplified, what is the most difficult operation? There are no exponents now, so division is the most difficult operation.

$$\color{black}{12 \div 4} + 5 - 3$$

$$12 \div 4=3$$

3. What is the next most difficult operation? Addition and subtraction are the same level of difficulty and I have to calculate them from left to right, so I will start with addition.

$$\color{black}{3+5}-3$$

$$3+5=8$$

Then I'll do subtraction.

$$8-3=5$$

## A Note on Addition and Subtraction

When you have addition and subtraction in the same problem, you have to be careful to make sure you do them in the right order or you may get totally different answers.

For example, if you had a problem like 7 - 2 + 4 and you did subtraction first, your answer would be 9.

7 - 2 + 4

5 + 4

9

7 - 2 + 4

7 - 6

1

The correct answer for the example above is 9.

Many teachers will tell you to calculate addition and subtraction from left to right and if you remember to follow this rule, you’ll always get the right answer.

However, there is another method that I think is much easier.

When I do addition and subtraction, I change all the subtraction terms to negatives and then I add everything together in whatever order is easiest.

## Change to Negatives Method

8 + 5 -7 -1 + 3 -2

becomes

8 + 5 + (-7) + (-1) + 3 + (-2)

I usually prefer to add up all my positive numbers first.

8 + 5 + 3 = 16

Then I add up all my negative numbers.

(-7) + (-1) + (-2) = (-10)

And finally, I add the positives and negatives together.

16 + (-10) = 6

I could simplify the exact same problem by calculating the addition and subtraction from left to right and I would get the same answer.

## Left to Right Method

8 + 5 -7 -1 + 3 - 2

Starting from the left and going in order...

8 + 5 = 13

13 - 7 = 6

6 - 1 = 5

5 + 3 = 8

8 - 2 =6

Notice that the final answer is the same as the answer we found using the "Change to Negatives" method.

It doesn't matter which method you use. Just use the one that makes the most sense to you :)

## A Note on Multiplication and Division

When you have multiplication and division in the same problem, it doesn’t matter which order you do them in because either order will give you the same answer.

For example, if you had $$3 \times 8 \div 2$$, you could do multiplication first:

$$\color{black}{3 \times 8} \div 2$$

$$3 \times 8 = 24$$

$$24 \div 2 = 12$$

Or you could do division first:

$$3 \times \color{black}{8 \div 2}$$

$$8\div 2 = 4$$

$$3 \times 4 = 12$$

Notice that you get the same answer either way.

In addition to the traditional grouping symbols (), [], {}, there are also a couple of symbols that imply grouping even if they don’t have parentheses around them.

$$\frac{4\times 5^{3} + 1}{6-3+2\times 9}$$ should be interpreted as $$\frac{(4\times 5^{3} + 1)}{(6-3+2\times 9)}$$

$$\sqrt{7+9\times 2}$$ should be interpreted as $$\sqrt{(7+9\times 2)}$$

## Fractions & Division Bars

Order of operations works the same for fractions as it does for normal numbers. If you need a refresher on how to add fractions, subtract fraction, multiply fractions, divide fractions, or raise a fraction to an exponent, check out the linked pages.

If you see a complicated fraction like this one $$\frac{3+5\times 6}{2\times 4 -5}$$, first treat the numerator (top) and denominator (bottom) as if they have grouping symbols around them.

Then, remember that the fraction bar represents division so you can divide the simplified numerator (top) by the simplified denominator (bottom).

## Online Practice

Order of Operations (No Exponents)

Order of Operations

Order of Operations with Negative Numbers

Order of Operations Challenge

## Printable Worksheets

CommonCoreSheets.com

This resource has 10 different worksheets with answer keys.