3.2 Grouping symbols and the order of operations

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:

( ), [ ], { },

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.

Practice set a

If possible, determine the value of each of the following.

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.

Practice set b

Determine the value of each of the following.

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.

$\begin{array}{cc}\hfill 3+5\cdot 2& =8\cdot 2\hfill \\ & =\mathrm{16}\hfill \end{array}$

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

$\begin{array}{cc}\hfill 3+5\cdot 2& =3+\mathrm{10}\hfill \\ & =\mathrm{13}\hfill \end{array}$

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.