1.2 Order of Operations (PEMDAS)

Learning Objectives

  1. Apply the correct order of operations (PEMDAS). (numeric only at this point).
  2. Recognize and avoid common pitfalls in order of operations.

Order of Operations (PEMDAS)

Suppose we want to simplify:

3 + 2 × 4

Some people might add first: (3 + 2) × 4 = 20.

Others might multiply first: 3 + (2 × 4) = 11.

Which is correct?

Mathematics requires precision. To avoid confusion, we use a shared set of rules called the order of operations. These rules tell us the order in which to simplify expressions so that everyone gets the same answer.

The order is:

  1. Parentheses (Grouping symbols): parentheses ( ), brackets [ ], braces { }, absolute value | |, and fraction bars.
  2. Exponents: powers and roots. Example: 5³ = 5 × 5 × 5.
  3. Multiplication and Division: from left to right.
  4. Addition and Subtraction: from left to right.

📌 Important Reminder: Multiplication does not always come before division. Addition does not always come before subtraction. These pairs are done in order from left to right.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.”

Table 1.2
Parentheses
Exponents
Multiplication and Division
Addition and Subtraction		 Please
Excuse
My Dear
Aunt Sally

It’s good that “My Dear” goes together, as this reminds us that multiplication and division have equal priority. We do them in order from left to right.

Similarly, “Aunt Sally” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

Rather than writing the order of operations as PEMDAS, it may be better to write it as:

Logo showing the letters PEMDAS for the order of operations. The P E appear on the left, while M A are stacked above D S on the right to show that multiplication and division, and addition and subtraction, have equal priority.
Figure 1.2

 

 

 

 

 

This emphasizes that Multiplication–Division and Addition–Subtraction are at the same level.

Understanding Exponents

Before applying order of operations, we need to understand exponents.

Definition

An exponential expression represents repeated multiplication:

5^3 = 5 \times 5 \times 5

Parts of an Exponential Expression

  • Base: the number being multiplied (5)
  • Exponent: how many times it is multiplied (3)

Example 1.2.pre 1

Evaluate: 2^4

Solution

2^4 = 2 \times 2 \times 2 \times 2 = 16

 

Example 1.2.pre 2

Evaluate: 10^3

Solution

10^3 = 10 \times 10 \times 10 = 1000

 

Special Cases

a. 5^1 = 5

b. 7^0 = 1

We will discuss zero exponents in a later lesson.

 

 

Example 1.2.1

Simplify: 5 + 2 × 3

Solution

Start with multiplication, then add.

5 + 2 × 3

5 + 6

11

Example 1.2.2

Simplify: 10 – 3 × 2 + 5

Solution

Start with multiplication, then add and subtract left to right.

10 – 3 × 2 + 5

10 – 6 + 5

4 + 5

9

 

📌 Note: Subtraction as “Adding the Opposite”
PEMDAS says to perform addition and subtraction from left to right. But here’s a helpful trick:

  • Subtraction can be rewritten as “adding the opposite.”
  • Example: 10 – 3 + 5 can be rewritten as 10 + (–3) + 5.

Once everything is written as addition, you can rearrange terms because addition is commutative (order doesn’t matter) and associative (grouping doesn’t matter).  The commutative property changes the order of numbers, while the associative property changes the grouping.

Both methods will lead to the same result:

  • Left to right: (10 – 3) + 5 = 7 + 5 = 12.
  • Rewrite: 10 + (–3) + 5 = (10 + 5) + (–3) = 15 – 3 = 12.

The key idea: If you keep the signs attached to the numbers, you can safely rearrange when adding.  These ideas will come up again when we solve equations.

 

Example 1.2.3

Simplify: 3 × 42

Solution

Start with exponents, then multiply.

3 × 42

3 × 16

48

Example 1.2.4

Simplify: 20 ÷ 10 × 2

Solution

Work left to right (division and multiplication have equal priority).

20 ÷ 10 × 2

2 × 2

4

Note: It is common for students to multiply first because it seems to come first in PEMDAS, but multiplication and division are done left to right.

Example 1.2.5

Simplify: (20 – 3 × 5)2

Solution

Start with parentheses. Inside the parentheses, multiply first, then subtract.

(20 – 3 × 5)2

(20 – 15)2

(5)2

25

Example 1.2.6

Evaluate: \frac{35 - 2(10 - 6)^2}{6 + 3^2}

Solution

📌 Reminder: A fraction bar acts as a grouping symbol. Simplify the numerator and denominator separately before dividing.

\frac{\left[35 - 2(10 - 6)^2\right]}{\left[6 + 3^2\right]}

In the numerator, start by evaluating what is in parentheses, then apply the exponent. In the denominator, begin with exponents.

\frac{\left[35 - 2(4)^2\right]}{\left[6 + 9\right]}

\frac{\left[35 - 2(16)\right]}{\left[15\right]}

\frac{\left[35 - 32\right]}{\left[15\right]}

\frac{3}{15} = \frac{1}{5}

Note: The brackets were only included to emphasize simplifying numerator and denominator separately.

Example 1.2.7

Evaluate: 250 - \left|80 - 10(5 - 2)^2\right|

Solution

Begin by working inside the absolute value. Within the absolute value, evaluate the parentheses first.

250 - \left|80 - 10(3)^2\right|

Next evaluate the exponent.

250 - \left|80 - 10\cdot9\right|

Next, multiply.

250 - \left|80 - 90\right|

Now subtract inside the absolute value.

250 - \left|-10\right|

Take the absolute value.

250 - 10

Finally, subtract:

250 - 10 = 240

 

Try It

Parentheses Matter

Be careful when working with negative numbers and exponents. Parentheses can change the meaning of an expression.

Key Idea:

  • (-4)^2 = 16
  • -4^2 = -16

📌 Important Reminder
Exponents are applied before a leading negative unless parentheses indicate otherwise.

Example 1.2.8

Evaluate: (-5)^2

Solution

(-5)^2 = (-5)\times(-5) = 25

 

Example 1.2.9

Evaluate: -5^2

Solution

-5^2 = -(5^2) = -25

 

Common Pitfalls

🚩 Pitfall 1: Thinking multiplication always comes before division.

  • Wrong: 20 ÷ 10 × 2 = 20 ÷ (10 × 2) = 1.
  • Correct: Work left to right → (20 ÷ 10) × 2 = 4.

🚩 Pitfall 2: Forgetting to apply exponents first.

  • Wrong: 3 × 4² = (3 × 4)² = 144.
  • Correct: 4² = 16 → 3 × 16 = 48.

🚩 Pitfall 3: Mishandling negatives with exponents.

  • (–3)² = (–3)(–3) = 9.
  • –3² = –(3²) = –9.

Try It

 

 

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