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:

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

Example 1: Evaluate  5² x 2⁴

Solution:

Example 2: Evaluate  289 – (3 x 5)²

Solution: 

Example 3: Evaluate  8 + (2 x 5) x 3⁴ ÷ 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 square-shaped room is 12 feet, then the area of the room is (12 feet)².

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.