1.3.4: Order of Operations
- Page ID
- 87275
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.3.4 Learning Objectives
- Use the order of operations to evaluate a numerical expression
- Use a calculator to evaluate a numerical expression
Grouping Symbols
Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations is to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:
( ), [ ], { }, \(\frac {\text{ }}{\text{ }}\), \(\sqrt {\text{ }}\)
Parentheses: ( )
Brackets: [ ]
Braces: { }
Fraction Bar: \(\frac {\text{ }}{\text{ }}\)
Radical: \(\sqrt {\text{ }}\)
In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.
Example 1
If possible, determine the value of each of the following.
\(9 + (3 \cdot 8) \nonumber\)
Solution
Since 3 and 8 are within parentheses, they are to be combined first.
\(\begin{array} {rcl} {9 + (3 \cdot 8)} & = & {9 + 24} \\ {} & = & {33} \end{array}\nonumber\)
Thus,
\[9 + (3 \cdot 8) = 33 \nonumber\]
Example 2
\((10 \div 0) \cdot 6 \nonumber\)
Solution
Since \(10 \div 0\) is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."
Multiple Grouping Symbols
When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.
Example 3
Determine the value of each of the following.
\(2 + (8 \cdot 3) - (5 + 6)\nonumber\)
Solution
Combine 8 and 3 first, then combine 5 and 6.
\(\begin{array} {ll} {2 + 24 - 11} & {\text{ Now combine left to right.}} \\ {26 - 11} & {} \\ {15} & {} \end{array}\nonumber\)
Example 4
\(10 + [30 - (2 \cdot 9)]\nonumber\)
Solution
Combine 2 and 9 since they occur in the innermost set of parentheses.
\(\begin{array} {ll} {10 + [30 - 18]} & {\text{ Now combine 30 and 18.}} \\ {10 + 12} & {} \\ {22} & {} \end{array}\nonumber\)
The Order of Operations
Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of \(3 + 5 \cdot 2\). We could do either of two things:
Add 3 and 5, then multiply this sum by 2.
\(\begin{array} {rcl} {3 + 5 \cdot 2} & = & {8 \cdot 2} \\ {} & = & {16} \end{array}\)
Multiply 5 and 2, then add 3 to this product.
\(\begin{array} {rcl} {3 + 5 \cdot 2} & = & {3 + 10} \\ {} & = & {13} \end{array}\)
We now have two different values for one expression. To determine the correct value, we must use the accepted order of operations.
Order of Operations
- Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4 described below,
- Perform all exponential and root operations.
- Perform all multiplications and divisions, moving left to right.
- Perform all additions and subtractions, moving left to right.
Example 5
Determine the value of each of the following.
\(\begin{array} {ll} {21 + 3 \cdot 12} & {\text{ Multiply first.}} \\ {21 + 36} & {\text{ Add.}} \\ {57} & {} \end{array}\)
Example 6
\(\begin{array} {ll} {(15 - 8) + 5 \cdot (6 + 4).} & {\text{ Simplify inside parentheses first.}} \\ {7 + 5 \cdot 10} & {\text{ Multiply.}} \\ {7 + 50} & {\text{ Add.}} \\ {57} & {} \end{array}\)
Example 7
\(\begin{array} {ll} {63 - (4 + 6 \cdot 3) + 76 - 4} & {\text{ Simplify first within the parenthesis by multiplying, then adding.}} \\ {63 - (4 + 18) + 76 - 4} & {} \\ {63 - 22 + 76 - 4} & {\text{ Now perform the additions and subtractions, moving left to right.}} \\ {41 + 76 - 4} & {\text{ Add 41 and 76: 41 + 76 = 117.}} \\ {117 - 4} & {\text{ Subtract 4 from 117: 117 - 4 = 113.}} \\ {113} & {} \end{array}\)
Example 8
\(\begin{array} {ll} {7 \cdot 6 - 4^2 + 1^5} & {\text{ Evaluate the exponential forms, moving left to right.}} \\ {7 \cdot 6 - 16 - 1} & {\text{ Multiply 7 and 6: 7 \cdot 6 = 42}} \\ {42 - 16 + 1} & {\text{ Subtract 16 from 42: 42 - 16 = 26}} \\ {26 + 1} & {\text{ Add 26 and 1: 26 + 1 = 27}} \\ {27} & {} \end{array}\)
Example 9
\(\begin{array} {ll} {6 \cdot (3^2 + 2^2) + 4^2} & {\text{ Evaluate the exponential forms in the parentheses: } 3^2 = 9 \text{ and } 2^2 = 4} \\ {6 \cdot (9 + 4) + 4^2} & {\text{ Add the 9 and 4 in the parentheses: 9 + 4 = 13}} \\ {6 \cdot (13) + 4^2} & {\text{ Evaluate the exponential form: } 4^2 = 16} \\ {6 \cdot (13) + 16} & {\text{ Multiply 6 and 13: } 6 \cdot 13 = 78} \\ {78 + 16} & {\text{ Add 78 and 16: 78 + 16 = 94}} \\ {94} & {} \end{array}\)
Example 10
\(\begin{array} {ll} {\dfrac{6^2 + 2^2}{4^2 + 6 \cdot 2^2} + \dfrac{1^2 + 8^2}{10^2 - 19 \cdot 5}} & {\text{ Recall that the bar is a grouping symbol.}} \\ {} & {\text{ The fraction } \dfrac{}{} \text{ is equivalent to } (6^2 + 2^2) \div (4^2 + 6 \cdot 2^2)} \\ {\dfrac{36 + 4}{16 + 6 \cdot 4} + \dfrac{1 + 64}{100 - 19 \cdot 5}} & {} \\ {\dfrac{36 + 4}{16 + 24} + \dfrac{1 + 64}{100 - 95}} & {} \\ {\dfrac{40}{40} + \dfrac{65}{5}} & {} \\ {1 + 13} & {} \\{14} & {} \end{array}\)