# 3.2 Grouping symbols and the order of operations

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses grouping symbols and the order of operations. By the end of the module students should be able to understand the use of grouping symbols, understand and be able to use the order of operations and use the calculator to determine the value of a numerical expression.

## Section overview

• Grouping Symbols
• Multiple Grouping Symbols
• The Order of Operations
• Calculators

## 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.

$9+\left(3\cdot 8\right)$

Since 3 and 8 are within parentheses, they are to be combined first.

$\begin{array}{cc}\hfill 9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\left(3\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}8\right)& =\phantom{\rule{2px}{0ex}}9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{24}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{33}\hfill \end{array}$

Thus,

$9+\left(3\cdot 8\right)=\text{33}$

$\left(\text{10}÷0\right)\cdot 6$

Since $\text{10}÷0$ 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.

$\text{16}-\left(3\cdot 2\right)$

10

$5+\left(7\cdot 9\right)$

68

$\left(4+8\right)\cdot 2$

24

$\text{28}÷\left(\text{18}-\text{11}\right)$

4

$\left(\text{33}÷3\right)-\text{11}$

0

$4+\left(0÷0\right)$

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 b

Determine the value of each of the following.

$2+\left(8\cdot 3\right)-\left(5+6\right)$

Combine 8 and 3 first, then combine 5 and 6.

$\begin{array}{cc}2+\mathrm{24}-\mathrm{11}\hfill & \text{Now combine left to right.}\\ \mathrm{26}-\mathrm{11}\hfill & \\ \mathrm{15}\hfill & \end{array}$

$\text{10}+\left[\text{30}-\left(2\cdot 9\right)\right]$

Combine 2 and 9 since they occur in the innermost set of parentheses.

$\begin{array}{cc}\mathrm{10}+\left[\mathrm{30}-\mathrm{18}\right]\hfill & \text{Now combine 30 and 18.}\\ \mathrm{10}+\mathrm{12}\hfill & \\ \mathrm{22}\hfill & \end{array}$

## Practice set b

Determine the value of each of the following.

$\left(\text{17}+8\right)+\left(9+\text{20}\right)$

54

$\left(\text{55}-6\right)-\left(\text{13}\cdot 2\right)$

23

$\text{23}+\left(\text{12}÷4\right)-\left(\text{11}\cdot 2\right)$

4

$\text{86}+\left[\text{14}÷\left(\text{10}-8\right)\right]$

93

$\text{31}+\left\{9+\left[1+\left(\text{35}-2\right)\right]\right\}$

74

${\left\{6-\left[\text{24}÷\left(4\cdot 2\right)\right]\right\}}^{3}$

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.

$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}8\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}2\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{16}\hfill \end{array}$

Multiply 5 and 2, then add 3 to this product.

$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}3\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{10}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{13}\hfill \end{array}$

We now have two values for one number. To determine the correct value, we must use the accepted order of operations .

## Order of operations

1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4 described below,
2. Perform all exponential and root operations.
3. Perform all multiplications and divisions, moving left to right.
4. Perform all additions and subtractions, moving left to right.

## Sample set c

Determine the value of each of the following.

$\begin{array}{cc}\mathrm{21}+3\cdot \mathrm{12}\hfill & \text{Multiply first.}\hfill \\ \mathrm{21}+\mathrm{36}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$

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

$\begin{array}{cc}\mathrm{63}-\left(4+6\cdot 3\right)+\mathrm{76}-4\hfill & \text{Simplify first within the parenthesis by multiplying, then adding.}\hfill \\ \mathrm{63}-\left(4+\mathrm{18}\right)+\mathrm{76}-4\hfill & \hfill \\ \mathrm{63}-\mathrm{22}+\mathrm{76}-4\hfill & \text{Now perform the additions and subtractions, moving left to right.}\hfill \\ \mathrm{41}+\mathrm{76}-4\hfill & \text{Add 41 and 76:}\phantom{\rule{8px}{0ex}}\mathrm{41}+\mathrm{76}=\mathrm{117}.\hfill \\ \mathrm{117}-4\hfill & \text{Subtract 4 from 117:}\phantom{\rule{8px}{0ex}}\mathrm{117}-4=\mathrm{113}.\hfill \\ \mathrm{113}\hfill & \end{array}$

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