**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 Part 1 (04:50 mins)__|__Transcript____Order of Operations Part 2 (01:07 mins)__|__Transcript____Examples of Order of Operations Part 1 (05:49 mins)__|__Transcript____Examples of Order of Operations Part 2 (04:21 mins)__|__Transcript__

**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.

Figure 1

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.

Figure 3

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?

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:**

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

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