What if your teacher asked you to evaluate the expression \begin{align*}3+2 \times 6 \div (3-1)\end{align*}? Which should you do first, the addition, subtraction, multiplication, or division? What should you do second, third, and fourth? Also, should the parentheses affect your decisions?

### PEMDAS

#### The Mystery of Math Verbs

Some math verbs are “stronger” than others and must be done first. This method is known as the **order of operations.**

A mnemonic (a saying that helps you remember something difficult) for the order of operations is PEMDAS: Please Excuse My Daring Aunt Sophie.

The order of operations:

Whatever is found inside **PARENTHESES** must be done first. **EXPONENTS** are to be simplified next. **MULTIPLICATION** and **DIVISION** are equally important and must be performed moving left to right. **ADDITION** and **SUBTRACTION** are also equally important and must be performed moving left to right.

#### Let's use the order of operations to simplify the following expressions:

- \begin{align*}(7-2) \times 4 \div 2-3.\end{align*}

First, we check for parentheses. Yes, there they are and must be done first.

\begin{align*}(7 - 2) \times 4 \div 2 - 3 = (5) \times 4 \div 2 - 3\end{align*}

Next we look for exponents (little numbers written a little above the others). No, there are no exponents so we skip to the next math verb.

Multiplication and division are equally important and must be done from left to right.

\begin{align*}5 \times 4 \div 2 - 3 & = 20 \div 2 - 3\\ 20 \div 2 - 3 & = 10 - 3\end{align*}

Finally, addition and subtraction are equally important and must be done from left to right.

\begin{align*}10-3 = 7\end{align*}

This is your answer.

- \begin{align*}3 \times 5-7 \div 2\end{align*}

There are no parentheses and no exponents. Go directly to multiplication and division from left to right: \begin{align*} 3 \times 5 - 7 \div 2 = 15 - 7 \div 2 = 15 - 3.5\end{align*}

Now subtract: \begin{align*}15 - 3.5 = 11.5\end{align*}

- \begin{align*}3 \times (5 - 7) \div 2\end{align*}

Parentheses must be done first: \begin{align*}3 \times (-2) \div 2\end{align*}

There are no exponents, so multiplication and division come next and are done left to right: \begin{align*}3 \times (-2) \div 2 = -6 \div 2 = -3\end{align*}

- \begin{align*}(3 \times 5) - (7 \div 2)\end{align*}

Parentheses must be done first: \begin{align*}(3 \times 5) - (7 \div 2) = 15 - 3.5\end{align*}

There are no exponents, multiplication, division, or addition, so simplify:

\begin{align*}15 - 3.5 = 11.5\end{align*}

#### Parentheses

Parentheses are used two ways. The first is to alter the order of operations in a given expression, such as in problem 3 and 4 from above. The second way is to clarify an expression, making it easier to understand.

Some expressions contain no parentheses, while others contain several sets of parentheses. Some expressions even have parentheses inside parentheses, called **nested parentheses**! Always start at the innermost parentheses and work outward.

#### Let's practice this and use the order of operations to evaluate the following expression when \begin{align*}x=2\end{align*} : \begin{align*}x=2\end{align*}

\begin{align*}(2x-3)+x^2-3\end{align*}

First, we will substitute in 2 for \begin{align*}x\end{align*}.

\begin{align*}(2(2)-3)+(2)^2-3\end{align*}

Now we will use the order of operations to evaluate the expression, starting inside the parentheses and then with the exponent.

\begin{align*}(2(2)-3)+(2)^2-3=(1)+4-3=2\end{align*}

We finish evaluating with addition and subtraction.

### Examples

#### Example 1

Earlier, you were asked to evaluate the expression \begin{align*}3+2 \times 6 \div (3-1)\end{align*}.

You cannot evaluate this expression in any order that you want. Instead, to evaluate this expression, you must use the order of operations (PEMDAS).

The first step is to simplify the expression inside the parentheses:

\begin{align*}3+2\times6\div (3-1)=3+2\times6\div 2\end{align*}

There are no exponents so the next step is multiplication and division from left to right:

\begin{align*}3+2\times6\div 2=3+12\div 2 \end{align*} \begin{align*}3+12\div 2=3+6\end{align*} Finally, add the two numbers together:

\begin{align*}3+6=9\end{align*} The expression is equal to 9.

#### Example 2

Use the order of operations to simplify \begin{align*}8-[19-(2+5)-7].\end{align*}

Begin with the innermost parentheses:

\begin{align*}8-[19-(2+5)-7]=8-[19-7-7]\end{align*}

Simplify according to the order of operations:

\begin{align*}8-[19-7-7]=8-[5]=3\end{align*}

#### Example 3

Use the order or operations to evaluate the following expression when \begin{align*}x=3 \text{ and } y=5:\end{align*}

\begin{align*}3\cdot y^2 -2(7-x)\end{align*}

First, we will substitute in 3 for \begin{align*}x\end{align*} and 5 for \begin{align*}y\end{align*}.

\begin{align*}3\cdot 5^2 -2(7-3)\end{align*}

Now we will use the order of operations to evaluate the expression, doing parentheses and exponents first, then multiplication, and finally subtraction.

\begin{align*}3\cdot 5^2 -2(7-3)=3\cdot 25-2(4)=75-8=67\end{align*}

Note that there was no division or addition, so we skipped those steps.

### Review

Use the order of operations to simplify the following expressions.

- \begin{align*}8 - (19 - (2 + 5) - 7)\end{align*}
- \begin{align*}2 + 7 \times 11 - 12 \div 3\end{align*}
- \begin{align*}(3 + 7) \div (7 - 12)\end{align*}
- \begin{align*}8 \cdot 5 + 6^2\end{align*}
- \begin{align*}9 \div 3 \times 7 - 2^3 + 7\end{align*}
- \begin{align*}8 + 12 \div 6 + 6\end{align*}
- \begin{align*}(7^2-3^2) \div 8\end{align*}

Evaluate the following expressions involving variables.

- \begin{align*}2y^2\end{align*} when \begin{align*}y = 5\end{align*}
- \begin{align*}3x^2 + 2x + 1\end{align*} when \begin{align*}x = 5\end{align*}
- \begin{align*}(y^2 - x)^2\end{align*} when \begin{align*}x = 2\end{align*} and \begin{align*}y = 1\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.3.