Parentheses in the Order of Operations

Did you ever go to the post office and send a package high-priority so that it’d arrive as soon as possible? Parentheses work just like that. Parentheses — ( ) — allow you to indicate that a piece of an expression is high-priority — that is, it has to be evaluated before the rest of the expression.

When an expression includes parentheses with only Big Four operators, just do the following:

1. Evaluate the contents of the parentheses.

2. Evaluate Big Four operators.

When an expression has more than one set of parentheses, don’t panic. Start by evaluating the contents of the first set and move left to right. Piece of cake!

Sample questions

1. Evaluate (6 – 2) + (15 / 3).

   9. Start by evaluating the contents of the first set of parentheses:

   (6 – 2) + (15 / 3) = 4 + (15 / 3)

   Move on to the next set of parentheses:

   = 4 + 5

   To finish up, evaluate the addition:

   = 9

2. Evaluate (6 + 1) x (5 – (–14) / –7).

   21. When a set of parentheses includes a mixed-operator expression, evaluate everything inside the parentheses according to the order of operations. Begin by evaluating the contents of the first set of parentheses: 6 + 1 = 7:

   (6 + 1) x (5 – (–14) / –7) =

   7 x (5 – (–14) / –7)

   Move to the next set of parentheses. This set contains a mixed-operator expression, so start with the division: –14 / –7 = 2:

   = 7 x (5 – 2)

   Complete the contents of the parentheses by evaluating the subtraction: 5 – 2 = 3:

   = 7 x 3

   Finish up by evaluating the multiplication: 7 x 3 = 21.

Practice questions

1. Evaluate 4 x (3 + 4) – (16 / 2).

2. What’s (5 + –8 / 2) + (3 x 6)?

3. Find (4 + 12 / 6 x 7) – (3 + 8).

4. (2 x –5) – (10 – 7) x (13 + –8) = ?

Following are the answers to the practice questions:

1. 4 x (3 + 4) – (16 / 2) = 20.

   Start by evaluating what’s inside the first set of parentheses:

   4 x (3 + 4) – (16 / 2)

   = 4 x 7 – (16 / 2)

   Next, evaluate the contents of the second set of parentheses:

   4 x 7 – 8

   Evaluate the multiplication and then the subtraction:

   = 28 – 8 = 20

2. (5 + –8 / 2) + (3 x 6) = 19.

   Inside the first set of parentheses, evaluate the division first and then the addition:

   (5 + –8 / 2) + (3 x 6)

   = (5 + –4) + (3 x 6)

   = 1 + (3 x 6)

   Next, evaluate the contents of the second set of parentheses:

   = 1 + 18

   Finish up by evaluating the addition:

   1 + 18 = 19

3. (4 + 12 / 6 x 7) – (3 + 8) = 7.

   Begin by focusing on the first set of parentheses, handling all multiplication and division from left to right:

   (4 + 12 / 6 x 7) – (3 + 8)

   = (4 + 2 x 7) – (3 + 8)

   = (4 + 14) – (3 + 8)

   Now do the addition inside the first set of parentheses:

   = 18 – (3 + 8)

   Next, evaluate the contents of the second set of parentheses:

   = 18 – 11

   Finish up by evaluating the subtraction:

   18 – 11 = 7

4. (2 x –5) – (10 – 7) x (13 + –8) = –25.

   Evaluate the first set of parentheses, then the second, and then the third:

   (2 x –5) – (10 – 7) x (13 + –8)

   = –10 – (10 – 7) x (13 + –8)

   = –10 – 3 x (13 + –8)

   = –10 – 3 x 5

   Next, do multiplication and then finish up with the subtraction:

   = –10 – 15 = –25