Order of Operations with Exponents

Unit 7 > Lesson 2 of 8

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
	Parentheses, Exponents, Multiplication & Division, Addition & Subtraction
	Note that although there are six words, they correspond to four rules.

Exercises

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:

Lessons on Pre-Algebra Order of Operations Order of Operations With Exponents Order of Operations with Integers Writing Algebraic Expressions Writing Algebraic Equations Practice Exercises Challenge Exercises Solutions Related Activities Interactive Puzzles Printable Worksheet