Order of Operations with Exponents

Problem: Evaluate this arithmetic expression: 18 + 36 ÷ 32

In the last lesson, we learned how to evaluate an arithmetic expression with more than one operation according to the following rules:

Rule 1: Simplify all operations inside parentheses.

Rule 2: Perform all multiplications and divisions, working from left to right.

Rule 3: Perform all additions and subtractions, working from left to right.

However, the problem above includes an exponent, so we cannot solve it without revising our rules.

Rule 1: Simplify all operations inside parentheses.

Rule 2: Simplify all exponents, working from left to right.

Rule 3: Perform all multiplications and divisions, working from left to right.

Rule 4: Perform all additions and subtractions, working from left to right.

We can solve the problem above using our revised order of operations.

Problem: Evaluate this arithmetic expression:  18 + 36 ÷ 32

Solution:

18 + 36 ÷ 32 = 18 + 36 ÷ 9 Simplify all exponents (Rule 2)
18 + 36 ÷ 9 = 18 + 4 Division (Rule 3)
18 + 4 = 22 Addition (Rule 4)

Let's look at some other examples that involve our new rules for order of operations.

Example 1: Evaluate  52 x 24

Solution:

52 x 24 = 25 x 24 Simplify all exponents, working from left to right (Rule 2)
25 x 24 = 25 x 16
25 x 16 = 400 Multiplication (Rule 3)

Example 2: Evaluate  289 - (3 x 5)2

Solution: 

289 - (3 x 5)2 = 289 - 152 Simplify all operations inside parentheses (Rule 1)
289 - 152 = 289 - 225 Simplify all exponents (Rule 2)
289 - 225 = 64 Subtraction (Rule 4)

Example 3: Evaluate  8 + (2 x 5) x 34 ÷ 9

Solution:

8 + (2 x 5) x 34 ÷ 9 = 8 + 10 x 34 ÷ 9 Simplify all operations inside parentheses (Rule 1)
8 + 10 x 34 ÷ 9 = 8 + 10 x 81 ÷ 9 Simplify all exponents (Rule 2)
8 + 10 x 81 ÷ 9 = 8 + 810 ÷ 9 Perform all multiplications and divisions, working from left to right (Rule 3)
8 + 810 ÷ 9 = 8 + 90
8 + 90 = 98 Addition (Rule 4)

Example 4: An interior decorator charges $15 per square foot to lay a carpet, and an installation fee of $150. If the room is square and each side measures 12 feet, how much will it cost to carpet it?

Solution: If one side of the square-shaped room is 12 feet, then the area of the room is (12 feet)2.

15 x 122 + 150 = 15 x 144 + 150 Simplify all exponents (Rule 2)
15 x 144 + 150 = 2,160 + 150 Multiplication (Rule 3)
2,160 + 150 = 2,310 Addition (Rule 4)

Answer: It will cost $2,310 to carpet this room.


Summary: 

To help us remember the order of operations, we can use the mnemonic PEMDAS, which stands for:

Please Excuse My Dear Aunt Sally

aunt_sally.gif

Parentheses, Exponents, Multiplication & Division, Addition & Subtraction

Note that although there are six words, they correspond to four rules.


Exercises

Directions: Complete each exercise by applying the rules for order of operations. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

1. 32 x 43
ANSWER BOX:   

RESULTS BOX: 

2. 27 - 256 ÷ 43
ANSWER BOX:   

RESULTS BOX: 

3. 9 x (5 + 3)2 - 144
ANSWER BOX:   

RESULTS BOX: 

4. 7 + 3 x 24 ÷ 6
ANSWER BOX:   

RESULTS BOX: 

5. A carpenter charges $10 per square foot to lay a floor. If a square-shaped hallway is 6 feet along one side, and the customer has a coupon for $25 off the total, then how much will the floor cost?
ANSWER BOX: $   

RESULTS BOX: