Order of Operations with Exponents
Problem: Evaluate this arithmetic expression: 18 + 36 ÷ 3^{2}
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 ÷ 3^{2}
Solution:
18 + 36 ÷ 3^{2}  = 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 5^{2} x 2^{4}
Solution:
5^{2} x 2^{4}  = 25 x 2^{4}  Simplify all exponents, working from left to right (Rule 2) 
25 x 2^{4}  = 25 x 16  
25 x 16  = 400  Multiplication (Rule 3) 
Example 2: Evaluate 289 – (3 x 5)^{2}
Solution:

Example 3: Evaluate 8 + (2 x 5) x 3^{4} ÷ 9
Solution:

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 squareshaped room is 12 feet, then the area of the room is (12 feet)^{2}.
15 x 12^{2} + 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
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.  3^{2} x 4^{3} 
2.  27 – 256 ÷ 4^{3} 
3.  9 x (5 + 3)^{2} – 144 
4.  7 + 3 x 2^{4} ÷ 6 
5.  A carpenter charges $10 per square foot to lay a floor. If a squareshaped 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? 