3.2: Grouping Symbols and the Order of Operations
- understand the use of grouping symbols
- understand and be able to use the order of operations
- use the calculator to determine the value of a numerical expression
Grouping Symbols
Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:
( ), [ ], { },
Parentheses
: ( )
Brackets
: [ ]
Braces
: { }
Bar
:
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.
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,
\[9 + (3 \cdot 8) = 33 \nonumber\]
\[(10 \div 0) \cdot 6\nonumber\]
Solution
Since \(10 \div 0\) is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."
Practice Set A
If possible, determine the value of each of the following.
\(16 - (3 \cdot 2)\)
- Answer
-
10
Practice Set A
\(5 + (7 \cdot 9)\)
- Answer
-
68
Practice Set A
\((4 + 8) \cdot 2\)
- Answer
-
24
Practice Set A
\(28 \div (18 - 11)\)
- Answer
-
4
Practice Set A
\((33 \div 3) - 11\)
- Answer
-
0
Practice Set A
\(4 + (0 \div 0)\)
- Answer
-
not possible (indeterminant)
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.
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.
\[\begin{array} {ll} {2 + 24 - 11} & {\text{ Now combine left to right.}} \\ {26 - 11} & {} \\ {15} & {} \end{array}\nonumber\]
\[10 + [30 - (2 \cdot 9)]\nonumber\]
Solution
Combine 2 and 9 since they occur in the innermost set of parentheses.
\[\begin{array} {ll} {10 + [30 - 18]} & {\text{ Now combine 30 and 18.}} \\ {10 + 12} & {} \\ {22} & {} \end{array}\nonumber\]
Practice Set B
Determine the value of each of the following.
\((17 + 8) + (9 + 20)\)
- Answer
-
54
Practice Set B
\((55 - 6) - (13 \cdot 2)\)
- Answer
-
23
Practice Set B
\(23 + (12 \cdot 4) - (11 \cdot 2)\)
- Answer
-
4
Practice Set B
\(86 + [14 \div (10 - 8)]\)
- Answer
-
93
Practice Set B
\(31 + \{9 + [1 + (35 - 2)]\}\)
- Answer
-
74
Practice Set B
\(\{6 - [24 \div (4 \cdot 2)]\}^3\)
- Answer
-
27
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 values for one number. 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.
Determine the value of each of the following.
\(\begin{array} {ll} {21 + 3 \cdot 12} & {\text{ Multiply first.}} \\ {21 + 36} & {\text{ Add.}} \\ {57} & {} \end{array}\)
\(\begin{array} {ll} {(15 - 8) + 5 \cdot (6 + 4).} & {\text{ Simplify inside parentheses first.}} \\ {7 + 5 \cdot 10} & {\text{ Multiply.}} \\ {7 + 50} & {\text{ Add.}} \\ {57} & {} \end{array}\)
\(\begin{array} {ll} {63 - (4 + 6 \cdot 3) + 76 - 4} & {\text{ Simplify first within the parenthesis by multiplying, then adding.}} \\ {63 - (4 + 18) + 76 - 4} & {} \\ {63 - 22 + 76 - 4} & {\text{ Now perform the additions and subtractions, moving left to right.}} \\ {41 + 76 - 4} & {\text{ Add 41 and 76: 41 + 76 = 117.}} \\ {117 - 4} & {\text{ Subtract 4 from 117: 117 - 4 = 113.}} \\ {113} & {} \end{array}\)
\(\begin{array} {ll} {7 \cdot 6 - 4^2 + 1^5} & {\text{ Evaluate the exponential forms, moving left to right.}} \\ {7 \cdot 6 - 16 - 1} & {\text{ Multiply 7 and 6: 7 \cdot 6 = 42}} \\ {42 - 16 + 1} & {\text{ Subtract 16 from 42: 42 - 16 = 26}} \\ {26 + 1} & {\text{ Add 26 and 1: 26 + 1 = 27}} \\ {27} & {} \end{array}\)
\(\begin{array} {ll} {6 \cdot (3^2 + 2^2) + 4^2} & {\text{ Evaluate the exponential forms in the parentheses: } 3^2 = 9 \text{ and } 2^2 = 4} \\ {6 \cdot (9 + 4) + 4^2} & {\text{ Add the 9 and 4 in the parentheses: 9 + 4 = 13}} \\ {6 \cdot (13) + 4^2} & {\text{ Evaluate the exponential form: } 4^2 = 16} \\ {6 \cdot (13) + 16} & {\text{ Multiply 6 and 13: } 6 \cdot 13 = 78} \\ {78 + 16} & {\text{ Add 78 and 16: 78 + 16 = 94}} \\ {94} & {} \end{array}\)
\(\begin{array} {ll} {\dfrac{6^2 + 2^2}{4^2 + 6 \cdot 2^2} + \dfrac{1^2 + 8^2}{10^2 - 19 \cdot 5}} & {\text{ Recall that the bar is a grouping symbol.}} \\ {} & {\text{ The fraction } \dfrac{}{} \text{ is equivalent to } (6^2 + 2^2) \div (4^2 + 6 \cdot 2^2)} \\ {\dfrac{36 + 4}{16 + 6 \cdot 4} + \dfrac{1 + 64}{100 - 19 \cdot 5}} & {} \\ {\dfrac{36 + 4}{16 + 24} + \dfrac{1 + 64}{100 - 95}} & {} \\ {\dfrac{40}{40} + \dfrac{65}{5}} & {} \\ {1 + 13} & {} \\{14} & {} \end{array}\)
Practice Set C
Determine the value of each of the following.
\(8 + (32 - 7)\)
- Answer
-
33
Practice Set C
\((34 + 18 - 2 \cdot 3) + 11\)
- Answer
-
57
Practice Set C
\(8(10) + 4(2 + 3) - (20 + 3 \cdot 15 + 40 - 5)\)
- Answer
-
0
Practice Set C
\(5 \cdot 8 + 4^2 - 2^2\)
- Answer
-
52
Practice Set C
\(4(6^2 - 3^3) \div (4^2 - 4)\)
- Answer
-
3
Practice Set C
\((8 + 9 \cdot 3) \div 7 + 5 \cdot (8 \div 4 + 7 + 3 \cdot 5)\)
- Answer
-
125
Practice Set C
\(\dfrac{3^3 - 2^3}{6^2 - 29} + 5 (\dfrac{8^2 + 2^4}{7^2 - 3^2}) \div \dfrac{8 \cdot 3 + 1^8}{2^3 - 3}\)
- Answer
-
7
Calculators
Using a calculator is helpful for simplifying computations that involve large numbers.
Use a calculator to determine each value.
\(9,842 + 56 \cdot 85\)
Solution
| key | Display Reads | ||
| Perform the multiplication first. | Type | 56 | 56 |
| Press | \(\times\) | 56 | |
| Type | 85 | 85 | |
| Now perform the addition. | Press | + | 4760 |
| Type | 9842 | 9842 | |
| Press | = | 14602 |
The display now reads 14,602.
\(42(27 + 18) + 105(810 \div 18)\)
Solution
| key | Display Reads | ||
| Operate inside the parentheses | Type | 27 | 27 |
| Press | + | 27 | |
| Type | 18 | 18 | |
| Press | = | 45 | |
| Multiply by 42. | Press | \(\times\) | 45 |
| Type | 42 | 42 | |
| Press | = | 1890 |
Place this result into memory by pressing the memory key.
| Key | Display Reads | ||
| Now operate in the other parentheses. | Type | 810 | 810 |
| Press | \(\div\) | 810 | |
| Type | 18 | 18 | |
| Press | = | 45 | |
| Now multiply by 105. | Press | \(\times\) | 45 |
| Type | 105 | 105 | |
| Press | = | 4725 | |
| We are now ready to add these two quantities together. | Press | + | 4725 |
| Press the memory recall key. | 1890 | ||
| Press | = | 6615 |
Thus, \(42(27 + 18) + 105 (810 \div 18) = 6,615\)
\(16^4 + 37^3\)
Solution
| Key | Display Reads | |
|---|---|---|
| Type | 16 | 16 |
| Press | \(\times\) | 16 |
| Type | 16 | 16 |
| Press | \(\times\) | 256 |
| Type | 16 | 16 |
| Press | \(\times\) | 4096 |
| Type | 16 | 16 |
| Press | = | 65536 |
| Press the memory key | ||
| Type | 37 | 37 |
| Press | \(\times\) | 37 |
| Type | 37 | 37 |
| Press | \(\times\) | 1396 |
| Type | 37 | 37 |
| Press | \(\times\) | 50653 |
| Press | + | 50653 |
| Press memory recall key | 65536 | |
| Press | = | 116189 |
| Key | Display Reads | |
|---|---|---|
| Type | 16 | 16 |
| Press | \(y^x\) | 16 |
| Type | 4 | 4 |
| Press | = | 4096 |
| Press | + | 4096 |
| Type | 37 | 37 |
| Press | \(y^x\) | 37 |
| Type | 3 | 3 |
| Press | = | 116189 |
Thus, \(16^4 + 37^3 = 116,189\)
We can certainly see that the more powerful calculator simplifies computations.
Nonscientific calculators are unable to handle calculations involving very large numbers.
\(85612 \cdot 21065\)
Solution
| Key | Display Reads | |
|---|---|---|
| Type | 85612 | 85612 |
| Press | \(\times\) | 85612 |
| Type | 21065 | 21065 |
| Press | = |
This number is too big for the display of some calculators and we'll probably get some kind of error message. On some scientific calculators such large numbers are coped with by placing them in a form called "scientific notation." Others can do the multiplication directly. (1803416780)
Practice Set D
\(9,285 + 86(49)\)
- Answer
-
13,499
Practice Set D
\(55(84 - 26) + 120 (512 - 488)\)
- Answer
-
6,070
Practice Set D
\(106^3 - 17^4\)
- Answer
-
1,107,495
Practice Set D
\(6,053^3\)
- Answer
-
This number is too big for a nonscientific calculator. A scientific calculator will probably give you \(2.217747109 \times 10^{11}\)
Exercises
For the following problems, find each value. Check each result with a calculator.
Exercise \(\PageIndex{1}\)
\(2 + 3 \cdot (8)\)
- Answer
-
26
Exercise \(\PageIndex{2}\)
\(18 + 7 \cdot (4 - 1)\)
Exercise \(\PageIndex{3}\)
\(3 + 8 \cdot (6 - 2) + 11\)
- Answer
-
46
Exercise \(\PageIndex{4}\)
\(1 - 5 \cdot (8 - 8)\)
Exercise \(\PageIndex{5}\)
\(37 - 1 \cdot 6^2\)
- Answer
-
1
Exercise \(\PageIndex{6}\)
\(98 \div 2 \div 7^2\)
Exercise \(\PageIndex{7}\)
\((4^2 - 2 \cdot 4) - 2^3\)
- Answer
-
0
Exercise \(\PageIndex{8}\)
\(\sqrt{9} + 14\)
Exercise \(\PageIndex{9}\)
\(\sqrt{100} + \sqrt{81} - 4^2\)
- Answer
-
3
Exercise \(\PageIndex{10}\)
\(\sqrt[3]{8} + 8 - 2\cdot 5\)
Exercise \(\PageIndex{11}\)
\(\sqrt[4]{16} - 1 + 5^2\)
- Answer
-
26
Exercise \(\PageIndex{12}\)
\(61 - 22 + 4[3 \cdot (10) + 11]\)
Exercise \(\PageIndex{13}\)
\(121 - 4 \cdot [(4) \cdot (5) - 12] + \dfrac{16}{2}\)
- Answer
-
97
Exercise \(\PageIndex{14}\)
\(\dfrac{(1 + 16) - 3}{7} + 5 \cdot (12)\)
Exercise \(\PageIndex{15}\)
\(\dfrac{8 \cdot (6 + 20)}{8} + \dfrac{3 \cdot (6 + 16)}{22}\)
- Answer
-
29
Exercise \(\PageIndex{16}\)
\(10 \cdot [8 + 2 \cdot (6 + 7)]\)
Exercise \(\PageIndex{17}\)
\(21 \div 7 \div 3\)
- Answer
-
1
Exercise \(\PageIndex{18}\)
\(10^2 \cdot 3 \div 5^2 \cdot 3 - 2 \cdot 3\)
Exercise \(\PageIndex{19}\)
\(85 \div 5 \cdot 5 - 85\)
- Answer
-
0
Exercise \(\PageIndex{20}\)
\(\dfrac{51}{17} + 7 - 2 \cdot 5 \cdot (\dfrac{12}{3})\)
Exercise \(\PageIndex{21}\)
\(2^2 \cdot 3 + 2^3 \cdot (6 - 2) - (3 + 17) + 11(6)\)
- Answer
-
90
Exercise \(\PageIndex{22}\)
\(26 - 2 \cdot \{\dfrac{6 + 20}{13} \}\)
Exercise \(\PageIndex{23}\)
\(2 \cdot \{(7 + 7) + 6 \cdot [4 \cdot (8 + 2)]\}\)
- Answer
-
508
Exercise \(\PageIndex{24}\)
\(0 + 10(0) + 15 \cdot \{4 \cdot 3 + 1\}\)
Exercise \(\PageIndex{25}\)
\(18 + \dfrac{7 + 2}{9}\)
- Answer
-
19
Exercise \(\PageIndex{26}\)
\((4 + 7) \cdot (8 - 3)\)
Exercise \(\PageIndex{27}\)
\((6 + 8) \cdot (5 + 2 - 4)\)
- Answer
-
144
Exercise \(\PageIndex{28}\)
\((21 - 3) \cdot (6 - 1) \cdot (7) + 4(6 + 3)\)
Exercise \(\PageIndex{29}\)
\((10 + 5) \cdot (10 + 5) - 4 \cdot (60 - 4)\)
- Answer
-
1
Exercise \(\PageIndex{30}\)
\(6 \cdot \{2 \cdot 8 + 3\} - (5) \cdot (2) + \dfrac{8}{4} + (1 + 8) \cdot (1 + 11)\)
Exercise \(\PageIndex{31}\)
\(2^5 + 3 \cdot (8 + 1)\)
- Answer
-
52
Exercise \(\PageIndex{32}\)
\(3^4 + 2^4 \cdot (1 + 5)\)
Exercise \(\PageIndex{33}\)
\(1^6 + 0^8 + 5^2 \cdot (2 + 8)^3\)
- Answer
-
25,001
Exercise \(\PageIndex{34}\)
\((7) \cdot (16) - 3^4 + 2^2 \cdot (1^7 + 3^2)\)
Exercise \(\PageIndex{35}\)
\(\dfrac{2^3 - 7}{5^2}\)
- Answer
-
\(\dfrac{1}{25}\)
Exercise \(\PageIndex{36}\)
\(\dfrac{(1 + 6)^2 + 2}{3 \cdot 6 + 1}\)
Exercise \(\PageIndex{37}\)
\(\dfrac{6^2 - 1}{2^3 - 3} + \dfrac{4^3 + 2 \cdot 3}{2 \cdot 5}\)
- Answer
-
14
Exercise \(\PageIndex{38}\)
\(\dfrac{5(8^2 - 9 \cdot 6)}{2^5 - 7} + \dfrac{7^2 - 4^2}{5 \cdot 5^2}\)
Exercise \(\PageIndex{39}\)
\(\dfrac{(2 + 1)^3 + 2^3 + 1^{10}}{6^2} - \dfrac{15^2 - [2 \cdot 5]^2}{5 \cdot 5^2}\)
- Answer
-
0
Exercise \(\PageIndex{40}\)
\(\dfrac{6^3 - 2 \cdot 10^2}{2^2} + \dfrac{18(2^3 + 7^2)}{2(19) - 3^3}\)
Exercise \(\PageIndex{41}\)
\(2 \cdot \{6 + [10^2 - 6\sqrt{25}]\}\)
- Answer
-
152
Exercise \(\PageIndex{42}\)
\(181 - 3 \cdot (2\sqrt{36} + 3 \sqrt[3]{64})\)
Exercise \(\PageIndex{43}\)
\(\dfrac{2 \cdot (\sqrt{81} - \sqrt[3] {125})}{4^2 - 10 + 2^2}\)
- Answer
-
\(\dfrac{4}{5}\)
Exercises for Review
Exercise \(\PageIndex{44}\)
The fact that 0 + any whole number = that particular whole number is an example of which property of addition?
Exercise \(\PageIndex{45}\)
Find the product \(4,271 \times 630\).
- Answer
-
2,690,730
Exercise \(\PageIndex{46}\)
In the statement \(27 \div 3 = 9\), what name is given to the result 9?
Exercise \(\PageIndex{47}\)
What number is the multiplicative identity?
- Answer
-
1
Exercise \(\PageIndex{48}\)
Find the value of \(2^4\).