3.2: Grouping Symbols and the Order of Operations

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Learning Objectives

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.

Sample Set A

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,

\[9 + (3 \cdot 8) = 33 \nonumber\]

Sample Set A

\[(10 \div 0) \cdot 6\nonumber\]

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.

Sample Set A

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.

\[\begin{array} {ll} {2 + 24 - 11} & {\text{ Now combine left to right.}} \\ {26 - 11} & {} \\ {15} & {} \end{array}\nonumber\]

Sample Set A

\[10 + [30 - (2 \cdot 9)]\nonumber\]

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.

Sample Set C

Determine the value of each of the following.

\(\begin{array} {ll} {21 + 3 \cdot 12} & {\text{ Multiply first.}} \\ {21 + 36} & {\text{ Add.}} \\ {57} & {} \end{array}\)

Sample Set C

\(\begin{array} {ll} {(15 - 8) + 5 \cdot (6 + 4).} & {\text{ Simplify inside parentheses first.}} \\ {7 + 5 \cdot 10} & {\text{ Multiply.}} \\ {7 + 50} & {\text{ Add.}} \\ {57} & {} \end{array}\)

Sample Set C

\(\begin{array} {ll} {63 - (4 + 6 \cdot 3) + 76 - 4} & {\text{ Simplify first within the parenthesis by multiplying, then adding.}} \\ {63 - (4 + 18) + 76 - 4} & {} \\ {63 - 22 + 76 - 4} & {\text{ Now perform the additions and subtractions, moving left to right.}} \\ {41 + 76 - 4} & {\text{ Add 41 and 76: 41 + 76 = 117.}} \\ {117 - 4} & {\text{ Subtract 4 from 117: 117 - 4 = 113.}} \\ {113} & {} \end{array}\)

Sample Set C

\(\begin{array} {ll} {7 \cdot 6 - 4^2 + 1^5} & {\text{ Evaluate the exponential forms, moving left to right.}} \\ {7 \cdot 6 - 16 - 1} & {\text{ Multiply 7 and 6: 7 \cdot 6 = 42}} \\ {42 - 16 + 1} & {\text{ Subtract 16 from 42: 42 - 16 = 26}} \\ {26 + 1} & {\text{ Add 26 and 1: 26 + 1 = 27}} \\ {27} & {} \end{array}\)

Sample Set C

\(\begin{array} {ll} {6 \cdot (3^2 + 2^2) + 4^2} & {\text{ Evaluate the exponential forms in the parentheses: } 3^2 = 9 \text{ and } 2^2 = 4} \\ {6 \cdot (9 + 4) + 4^2} & {\text{ Add the 9 and 4 in the parentheses: 9 + 4 = 13}} \\ {6 \cdot (13) + 4^2} & {\text{ Evaluate the exponential form: } 4^2 = 16} \\ {6 \cdot (13) + 16} & {\text{ Multiply 6 and 13: } 6 \cdot 13 = 78} \\ {78 + 16} & {\text{ Add 78 and 16: 78 + 16 = 94}} \\ {94} & {} \end{array}\)

Sample Set C

\(\begin{array} {ll} {\dfrac{6^2 + 2^2}{4^2 + 6 \cdot 2^2} + \dfrac{1^2 + 8^2}{10^2 - 19 \cdot 5}} & {\text{ Recall that the bar is a grouping symbol.}} \\ {} & {\text{ The fraction } \dfrac{}{} \text{ is equivalent to } (6^2 + 2^2) \div (4^2 + 6 \cdot 2^2)} \\ {\dfrac{36 + 4}{16 + 6 \cdot 4} + \dfrac{1 + 64}{100 - 19 \cdot 5}} & {} \\ {\dfrac{36 + 4}{16 + 24} + \dfrac{1 + 64}{100 - 95}} & {} \\ {\dfrac{40}{40} + \dfrac{65}{5}} & {} \\ {1 + 13} & {} \\{14} & {} \end{array}\)

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.

Sample Set D

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.

Sample Set D

\(42(27 + 18) + 105(810 \div 18)\)

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, \(42(27 + 18) + 105 (810 \div 18) = 6,615\)

Sample Set D

\(16^4 + 37^3\)

Solution

Nonscientific Calculators
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

Calculators with \(y^x\) Key
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.

Sample Set D

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 \(2^4\).