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Order of Operations
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Introduction

In this lesson, you will learn more about the order of operations and practice using it to solve complex problems.

Tip

A good idea when working with many operations at once 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.


These videos illustrate the lesson material below. Watching the videos is optional.


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.

In the same way that any language has grammar rules, math also has rules that must be followed. The order of operations is a list of rules that tells you which operations to perform first to get the correct answer. It helps you know how to interpret an equation so it means what it is supposed to mean.

Example 1
\begin{align*} 3 + 2 \times 4 \end{align*}

There are two possible interpretations for solving the problem above. One way is to add 3 and 2 together and then multiply the solution by 4. The other interpretation is to multiply 2 and 4 together first and then add 3 to the solution.

Consider what answers you get for both of the interpretations. In the first scenario, you get a set of 5 objects from the initial addition, and then you multiply by 4. This equals a total of 20 objects.

five circles where three of them are colored orange, and two of them are colored pink.

Figure 1

Twelve orange circles on the left, and eight pink circles on the right, which totals to 20.

Figure 2

In the other scenario, you get 8 from the initial multiplication, and then you add 3. This equals a total of 11 objects.

four pairs of pink circles which illustrates the equation 2 x 4 = 8.

Figure 3

three orange circles on the left and four pairs of pink circles on the right which illustrates the equation 3 + 8 = 11.

Figure 4

11 and 20 are significantly different answers. Which one is correct? Thankfully, the order of operations tells us how to find the answer.

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 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 inside of 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, if there is more than one multiplication and/or division operation, complete them in the same step, one at a time, in order from left to right. 
  • Addition and Subtraction:
    Addition and subtraction also work together. You can do subtraction or addition first because they are part of the same step. However, they must be completed in order from left to right across the equation.

"PEMDAS" is a frequently used expression in English to help students remember the order of operations:

  • P: for parentheses
  • E: for exponents
  • M: for multiplication
  • D: for division
  • A: for addition
  • S: for subtraction

Some people like the phrase “Please excuse my dear Aunt Sally” to help them remember the acronym.

Critical Thinking Challenge

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

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 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 & 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.

Consider Example 1. Which answer is correct? If you follow the order of operations, you will get 11 as the correct answer:
\begin{align*} &3 + 2 \times 4 &\color{red}\small\text{Simplify}\\\\ &3 + 8 &\color{red}\small\text{Multiply first}\\\\ &11 &\color{red}\small\text{Add last}\\\\ \end{align*}

What if you planned on 3 and 2 being added together first? How could you convey that to whoever was performing the equation? You would use parentheses: \((3+2) \times 4 = 20\). Parentheses tell you to perform that part of an equation first, regardless of what operation is inside of it.

Practicing the Order of Operations

Remember: It is acceptable and even encouraged to do these equations in steps and to write out each step.

Example 2
\(5 - 1 + 8\div4\)

\begin{align*}5 - 1 &+ 8\div4 &\color{red}\small\text{Simplify using order of operations} \\\\5 - 1 &+ 2 &\color{red}\small\text{Multiply or divide from left to right}\\\\4 &+ 2 &\color{red}\small\text{Add or subtract from left to right}\\\\&6 &\color{red}\small\text{Add or subtract from left to right} \\\end{align*}

As there are a few different addition and subtraction operations to complete, you will begin from the left and work your way over to the right. When you follow the order of operations, you find that the correct answer for this equation is 6.

Example 3
\((8 + 6\div3)\times 2\)
This equation shows that sometimes you need to use the order of operations within the order of operations.

\begin{align*}(8 + 6&\div3)\times 2 &\color{red}\small\text{Simplify using order of operations} \\\\(8 + 2&)\times 2 &\color{red}\small\text{Solve within the parentheses first} \\\\10&\times 2 &\color{red}\small\text{Solve within the parentheses}\\\\&20 &\color{red}\small\text{Multiply or divide from left to right} \\\end{align*}

Using the order of operations, you get 20 as the final answer.

Example 4
\(2(1+3)\)
This equation uses parentheses to indicate multiplication: \(2(1+3)\).

Note that all of the following expressions are appropriate ways to convey multiplication: \(2\times 4\), \(2(4)\), \((2)(4)\), and \(2\cdot 4\).

\begin{align*} &2(1+3) &\color{red}\small\text{Simplify using order of operations} \\\\ &2(4) &\color{red}\small\text{Solve within the parentheses first}\\\\&8 &\color{red}\small\text{Multiply or divide from left to right} \\\end{align*}

The final answer is 8.

Example 5
\(4 - 3^{2} + 2^{4}\div8\)
This equation contains exponents.

\begin{align*}4 - 3^{2} &+ 2^{4}\div8 &\color{red}\small\text{Simplify using order of operations} \\\\4 - 9 &+ 16\div8 &\color{red}\small\text{Solve the exponents}\\\\&4 - 9 + 2 &\color{red}\small\text{Multiply or divide from left to right}\\\\-5 &+ 2 &\color{red}\small\text{Add or subtract from left to right} \\\\ & -3 &\color{red}\small\text{Add or subtract from left to right} \end{align*}

The answer is -3, which is said as “negative three.”

Example 6
\((3(2+6))\div(10-4)\)
This equation uses multiple sets of parentheses.

\begin{align*}(3(2+6)&)\div(10-4) &\color{red}\small\text{Simplify using order of operations} \\\\(3(8)&)\div(6) &\color{red}\small\text{Solve within the parentheses first}\\\\24&\div6 &\color{red}\small\text{Multiply or divide from left to right}\\\\& 4&\color{red}\small\text{Multiply or divide from left to right} \\ \end{align*}

When there are multiple sets of parentheses in the same operation, start with the innermost set. Simplifying this problem using the order of operation will lead you to the answer of 4.

Remember to continue memorizing your single-digit multiplication. As you can see, multiplication is used a lot in these lessons and they will be easier if you know your multiplication.


Things to Remember


  • Simplify each math expression one step at a time. Use PEMDAS.
  • Rewrite each step after completing an operation. This ensures that no steps are forgotten or overlooked.
  • If there are more than one set of parentheses in the same operation, start with the inside parentheses.
  • Parentheses may be used to indicate multiplication.

Practice Problems

Evaluate the following expressions:
  1. \(0\div4\times7+5^{2}\div5\times5 = ?\) (
    Solution
    x
    Solution: 25
    Details:
    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 \(5 \times 5\). Replace the exponent with the answer 25.
    \(0 \div 4 \times 7 + 25 \div 5 \times 5\)

    Solve any multiplication or division from left to right. Divide 0 by 4.
    \(0 \div 4 \times 7 + 25 \div 5 \times 5\)

    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 \(0 \times 7\) and 25 divided by 5.
    \(0\times 7 + 25 \div 5 \times 5\)

    Replace \(0 \times 7\) and 25 divided by 5 with their respective answers, 0 and 5.
    \(0 + 5 \times 5\)

    Continue solving the multiplication of \(5 \times 5\).
    \(0 + 5 \times 5\)

    Replace the \(5 \times 5\) with the answer 25. The final step is to solve any addition.
    \(0 + 25\)

    Add \(0 + 25\).

    The answer is 25.
    )
  2. \(6 - 4 ^{2} \div 2 - 2^{3} + 3 = ?\) (
    Solution
    x
    Solution: \(-7\)
    Details:
    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 \(4 \times 4\) and \(2 × 2 × 2\). Replace the previous exponents in the problem with the answers 16 and 8 respectively.
    \(6 - 16 \div 2 - 8 + 3\)

    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 \(-7\).
    \(- 7\)
    )
  3. \(6 \div 1 - \lgroup 7 - 5 \rgroup \times 3^{2} \times 7 = ?\) (
    Video Solution
    x
    Solution: \(-120\)
    Details:

    (Order of Operations #3 (02:20 mins) | Transcript)
    )
  4. \(7 + 2 \times 3^{3} + 12 \div 2 = ?\) (
    Solution
    x
    Solution:
    67
    )
  5. \(2^{3} \times 5 \div \lgroup 5 - 1 \rgroup \div \lgroup 2 - 1 \rgroup \times 6 = ?\) (
    Video Solution
    )
  6. \(5^{3} - \lgroup 5 \times 2 \rgroup^{2} - 2^{4} - 2^{3} = ?\) (
    Solution
    x
    Solution:
    1
    )

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