Section 0.1: Order of Operations
- Page ID
- 192777
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We will rely heavily on these skills throughout this section.
- Evaluate \( 2^3+3^2 \)
- Evaluate \(3+ 4 \cdot 5\)
- Evaluate \(2\cdot 3 + 2 \cdot 2\)
Motivating Problem
A student simplified \(8+2\cdot5\) and got 50. Another student got 18. Who’s correct—and why?
Fun Fact
In the 1600s, some mathematicians used a vertical line, called a “vinculum” (similar to the one in a square root), to indicate grouping instead of parentheses. These early forms of notation eventually evolved into the clear order-of-operations conventions we use today.
The Goal
In this section, we’ll review how to correctly simplify expressions using the order of operations: grouping symbols, exponents, multiplication/division (left to right), and addition/subtraction (left to right). This is the foundation we need before working with algebraic expressions.
Simplify Expressions Using the Order of Operations
To simplify an expression means to perform all the possible mathematical operations. For example, to simplify \(4\cdot 2 + 1\) we’d first multiply \(4\cdot 2\) to get \(8\) and then add the \(1\) to get \(9\). A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
\[4\cdot 2 + 1\nonumber\]
\[8 + 1\nonumber\]
\[9\nonumber\]
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
To simplify an expression, do all operations in the expression.
You are familiar with most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings and result in different values. For example, consider the expression:
\[4 + 3\cdot 7\nonumber\]
If you simplify this expression, what do you get?
Some students incorrectly say \(49\) since \(4+3\) give \(7\) and \(7\cdot 7\) is \(49\).
\[4 + 3\cdot 7\nonumber\]
\[7 \cdot 7\nonumber\]
\[49\nonumber\]
Others say \(25\) since \(3\cdot 7\) is \(21\) and \(21 + 4\) makes \(25\).
\[4 + 3\cdot 7\nonumber\]
\[4 + 21\nonumber\]
\[25\nonumber\]
Imagine the confusion in our banking system if every problem had several different correct answers!
The same expression should give the same result, so mathematicians established some guidelines early on, which are called the Order of Operations.
- Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost grouping symbols first.
- Exponents
- Simplify all expressions with exponents.
- Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
- Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each keyword and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.”
\[\begin{align*} &\textbf{P}\text{arentheses} & & \textbf{P}\text{lease} \\[5pt]
&\textbf{E}\text{xponents} & & \textbf{E}\text{xcuse} \\[5pt]
&\textbf{M}\text{ultiplication}\space\textbf{D}\text{ivision} & & \textbf{M}\text{y}\space\textbf{D}\text{ear} \\[5pt]
&\textbf{A}\text{ddition}\space\textbf{S}\text{ubtraction} & & \textbf{A}\text{unt}\space\textbf{S}\text{ally} \end{align*}\]
It’s good that “\(\textbf{M}\text{y}\space\textbf{D}\text{ear}\)” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “\(\textbf{A}\text{unt}\space\textbf{S}\text{ally}\)” goes together, reminding us that addition and subtraction also have equal priority and are performed in order from left to right.
Let’s try an example.
Simplify:
- \(4 + 3\cdot 7\)
- \((4 + 3)\cdot 7\)
Solution
1.| \(4 + 3 \cdot 7\) | |
| Are there any parentheses? No. | |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply first. | \(4 + {\color{red}{3 \cdot 7}}\) |
| Add. | \(4+21\) |
| \(25\) |
2.
| \((4 + 3)\cdot 7\) | |
| Are there any parentheses? Yes. | \({\color{red}{(4 + 3)}}\cdot 7\) |
| Simplify inside the parentheses. | \(({\color{red}{7}})7\) |
| Are there any exponents? No. | |
| Is there any multiplication or division? Yes. | |
| Multiply. | \(49\) |
Simplify:
- \(8 + 3\cdot 9\)
- \((8 + 3)\cdot 9\)
- Answer
-
- \(35\)
- \(99\)
Simplify: \(18\div 6 + 4(5 - 2)\)
Solution
| Parentheses? Yes, subtract first. |
\(18\div 6 + 4(5 - 2)\) |
| Exponents? No. | |
| Multiplication or division? Yes. | \({\color{red}{18\div 6}} + {\color{red}{4(3)}}\) |
| Divide first because we multiply and divide left to right. | \(3+{\color{red}{4(3)}}\) |
| Any other multiplication or division? Yes. | |
| Multiply. | \(3 + 12\) |
| Any other multiplication or division? No. | |
| Any addition or subtraction? Yes. | \(15\) |
Simplify: \(30\div 5 + 10(3 - 2)\)
- Answer
-
\(16\)
When there are multiple grouping symbols, we simplify the innermost grouping symbols first and work outward.
Simplify: \(5 + 2^{3} + 3[6 - 3(4 - 2)]\).
Solution
| \(5 + 2^{3} + 3[6 - 3(4 - 2)]\) | |
| Are there any parentheses (or other grouping symbols)? Yes. | |
| Focus on the parentheses that are inside the brackets. | \(5 + 2^{3} + 3[6 - 3{\color{red}{(4 - 2)}}]\) |
| Subtract. | \(5 + 2^{3} + 3[6 - {\color{red}{3(2)}}]\) |
| Continue inside the brackets and multiply. | \(5 + 2^{3} + 3[{\color{red}{6 - 6}}]\) |
| Continue inside the brackets and subtract. | \(5 + 2^{3} + 3[{\color{red}{0}}]\) |
| The expression inside the brackets requires no further simplification. | |
| Are there any exponents? Yes. | \(5 + {\color{red}{2^{3}}}+ 3[0]\) |
| Simplify exponents. | \(5 + 8 + {\color{red}{3[0]}}\) |
| Is there any multiplication or division? Yes. | |
| Multiply. | \({\color{red}{5 + 8}}+0\) |
| Is there any addition or subtraction? Yes. | |
| Add. | \({\color{red}{13 + 0}}\) |
| Add. | \(13\) |
Simplify: \(7^{2} - 2[4(5 + 1)]\).
- Answer
-
\(1\)