The Order of Operations and Variables:

Order of Operations

Tip:

A good idea when working with many operations at a time is to do a little portion of the equation at a time, rewriting frequently. For example, do the portion within the parentheses and then rewrite the equation. Trying to do the entire equation at once can often lead to mistakes. Break it down into parts using the order of operations and do a little at a time.

What is the order of operations?

Operations are things like addition, subtraction, multiplication, and division. When you add two numbers together, you are performing the operation of addition on them. Similarly, when you multiply numbers together, you are performing the operation of multiplication.

The order of operations is the rule for what operations should be done first when there are several operations within the same equation.

The order of operations is like grammar rules for the language of math. It explains how to interpret an equation to mean what it is supposed to mean.

Applying the Order of Operations (PEMDAS)

The order of operations says that operations must be done in the following order: parentheses, exponents, multiplication, division, addition, and subtraction.

Parentheses

When there are parentheses, whatever is inside must be done first. The stuff inside the parentheses may also need to be broken down according to the order of operations as well. It is even possible to have parentheses within parentheses. In cases like this, work from the inside out.

Exponents

If there are exponents in the equation, these would be done next.

Multiplication and Division

Multiplication and division can be done together. In other words, it doesn’t matter if you do division or multiplication first, but they must be done after parentheses and exponents and before addition and subtraction.

Addition and Subtraction

Addition and subtraction also work together. You can do subtraction first, or you can do addition first. They are part of the same step, however, they can only be done after items in parentheses, exponents, and any multiplication and division.

PEMDAS

A frequently used expression in English to help students remember the order of operations is PEMDAS.

$This image displays the order of operations. At the top is a cartoon girl with arms raised in a questioning manner. In front of her is a banner that says, “What do I do first?” Below this banner is a box. Inside the box is a long mathematical expression using all the different operations. Seventy two plus eight raised to the third power multiplied to left parenthesis seven hundred eighty two multiplied by seventeen right parenthesis multiplied to the fraction four fifths. Below this expression is says, “When faced with a complex math question, we follow what is called the Order of Operations. Operations are things like addition, subtraction, multiplication and division. It is often helpful to think of the expression ‘PEMDAS’ to remember the correct order…” Below this box are four banners. Each banner has a letter or letters on the left side. These letters spell PEMDAS vertically. The explanation of what each letter stands for is written to the right. The first banner has a P on the left side. Within the banner is written parentheses (2+12). The next banner has an E on the left side. Within the banner is written Exponents. There is a six raised to the 2 power next to this word. The third banner has the letters M and D to the left. Inside the banner is written Multiply and Divide and the symbols for multiplication and division. The last banner has the letters A and S to the left side and the words Addition and subtraction with the symbols for addition and subtraction written inside.$

Another way to remember this is the phrase “Please Excuse My Dear Aunt Sally.”

Critical Thinking Challenge

Can you think of another phrase that could help you remember the order of operations?

Video Source (04:50 mins) | Transcript

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Remember to take it one step at a time and rewrite your equation after completing an operation. Doing this will help you keep track of what you’ve already done and make sure you don’t skip any steps.

Remember to continue to work on memorizing your single digit multiplication if they aren’t memorized yet. As you are beginning to see, we are using multiplication a lot in these lessons and they will be easier if you know your multiplication.

Additional Resources

Khan Academy: Intro to order of operations (09:39 mins, Transcript)
Khan Academy: Order of operations example (04:26 mins, Transcript)
Khan Academy: Order of operations (PEMDAS) (05:19 mins, Transcript)

Practice Problems

Evaluate the following expression:

$0\div4\times7+5^{2}\div5\times5 = ?$
$6 - 4 ^{2} \div 2 - 2^{3} + 3 = ?$
$6 \div 1 - \lgroup 7 - 5 \rgroup \times 3^{2} \times 7 = ?$
$7 + 2 \times 3^{3} + 12 \div 2 = ?$
$2^{3} \times 5 \div \lgroup 5 - 1 \rgroup \div \lgroup 2 - 1 \rgroup \times 6 = ?$
$5^{3} - \lgroup 5 \times 2 \rgroup^{2} - 2^{4} - 2^{3} = ?$

Solutions

25 (Written Solution)

There are various steps in the order of operations. Use only the steps required to solve this problem.

First, solve exponents. The 5 is the only number with an exponent.

$0 \div 4 \times 7 + 5^{2} \div 5 \times 5$

Solve the exponent by multiplying 5 × 5. Replace the exponent with the answer 25.

$0 \div 4 \times 7 + 25 \div 5 \times 5$

Now, solve any multiplication or division from left to right. Divide 0 by 4.

$0 \div 4 \times 7 + 25 \div 5 \times 5$

Now, replace the division of 0 by 4 with 0.

$0\times 7 + 25 \div 5 \times 5$

Solve any multiplication or division from left to right. Solve 0 × 7 and 25 divided by 5.

$0\times 7 + 25 \div 5 \times 5$

Replace 0 × 7 and 25 divided by 5 with their respective answers, 0 and 5.

$0 + 5 \times 5$

Continue solving the multiplication of 5 × 5.

$0 + 5 \times 5$

Replace the 5 × 5 with the answer 25. The final step is to solve any addition.

$0 + 25$

Add 0 + 25. The answer is 25.
$-7$ (Written Solution)

There are various steps in the Order of Operations. Use only the steps required to solve this particular problem.

First, resolve any exponents. There are two numbers with exponents, the 4 and 2.

$6 - 4^{2} \div 2 - 2^{3} + 3$

Solve the exponents by multiplying 4 × 4 and 2 × 2 × 2. Replace the previous exponents in the problem with the answers 16 and 8 respectively.

$6 - 16 \div 2 - 8 + 3$

Now, solve any multiplication or division from left to right. The division of 16 by 2 is the next step.

$6 - 16 \div 2 - 8 + 3$

The solution to the division of 16 by 2 is 8. Replace the division of 16 by 2 with the number 8.

$6 - 8 - 8 + 3$

The last step in the Order of Operations is to solve any addition or subtraction from left to right. Solve the subtraction 6 − 8.

$6 - 8 - 8 + 3$

The difference between 6 − 8 is $-2$. Replace the subtraction of 6 − 8 with a $-2$.

$- 2 - 8 + 3$

Now subtract $-2-8$.

$- 2 - 8 + 3$

The answer is $-10$. Replace the subtraction with $-10$. Proceed to the last step, add $-10$ and 3.

$- 10 + 3$

Replace $-10+3$ with the answer, which is $-7$.

$- 7$
$-120$ (Solution Video | Transcript)
67
60 (Solution Video | Transcript)
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