Order of Operations

Introduction to PEMDAS

The Order of Operations is a set of rules that defines the sequence in which mathematical operations should be performed in an expression. The acronym PEMDAS helps us remember the correct order:

PEMDAS stands for:

Parentheses (Brackets)
Exponents (Powers, Roots)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Note: Multiplication and Division have the same precedence, as do Addition and Subtraction. When operations of equal precedence appear, they are evaluated from left to right.

Comprehensive Examples

Basic Example

Expression: 2 + 3 × 4

Solution:

Multiplication before addition: 3 × 4 = 12
Then addition: 2 + 12 = 14

Result: 14

Parentheses Example

Expression: (2 + 3) × 4

Solution:

Parentheses first: (2 + 3) = 5
Then multiplication: 5 × 4 = 20

Result: 20

Exponents Example

Expression: 2 + 3² × 4

Solution:

Exponent first: 3² = 9
Multiplication before addition: 9 × 4 = 36
Then addition: 2 + 36 = 38

Result: 38

Division and Multiplication Example

Expression: 12 ÷ 4 × 3

Solution:

Division and multiplication are at the same level, so work from left to right:
12 ÷ 4 = 3
3 × 3 = 9

Result: 9

Addition and Subtraction Example

Expression: 10 - 5 + 2

Solution:

Addition and subtraction are at the same level, so work from left to right:
10 - 5 = 5
5 + 2 = 7

Result: 7

Complex Example

Expression: 2 × (3 + 4²) ÷ 5 - 6

Solution:

Inside parentheses: Exponent first: 4² = 16
Inside parentheses: Addition: 3 + 16 = 19
Multiplication: 2 × 19 = 38
Division: 38 ÷ 5 = 7.6
Subtraction: 7.6 - 6 = 1.6

Result: 1.6

Nested Parentheses Example

Expression: 5 + (2 × (8 - 3))

Solution:

Inner parentheses first: (8 - 3) = 5
Next level of parentheses: (2 × 5) = 10
Then addition: 5 + 10 = 15

Result: 15

Fraction Example

Expression: (3 + 5) ÷ (2 × 2)

Solution:

First parentheses: (3 + 5) = 8
Second parentheses: (2 × 2) = 4
Division: 8 ÷ 4 = 2

Result: 2

Negative Numbers Example

Expression: -2² + 5

Solution:

When a negative number is raised to a power, be careful with the negative sign
-2² means -(2²) = -(4) = -4 (the negative sign is outside the exponent)
Then addition: -4 + 5 = 1

Note: This is different from (-2)² = 4, where the negative number is inside parentheses before taking the exponent.

Result: 1

Decimal Example

Expression: 0.5 × (10 - 4) + 2.5

Solution:

Parentheses first: (10 - 4) = 6
Multiplication: 0.5 × 6 = 3
Addition: 3 + 2.5 = 5.5

Result: 5.5

Special Cases and Common Errors

Multiple Exponents

Expression: 2³²

Common Misconception: Calculate 2³ first, then square the result.

Correct Approach: In standard notation, exponents are evaluated from right to left when stacked.

This should be interpreted as 2^(3²) which is 2⁹ = 512

Result: 512

Implied Multiplication

Expression: 2(3 + 4)

Common Misconception: The parentheses might be confusing without the × symbol.

Correct Approach: Implied multiplication has the same precedence as explicit multiplication.

Parentheses first: (3 + 4) = 7
Implied multiplication: 2 × 7 = 14

Result: 14

Division with Fractions

Expression: 1 + 8 ÷ 2(4)

Common Misconception: Calculating 2(4) before division.

Correct Approach: Division and multiplication have the same precedence, so work left to right.

Division first: 8 ÷ 2 = 4
Multiplication: 4 × 4 = 16
Addition: 1 + 16 = 17

Result: 17

Different Methods for Solving Complex Expressions

Method 1: Step-by-Step PEMDAS

Expression: 3 + 4 × 2² - 8 ÷ 4

Exponents: 2² = 4
Multiplication and Division (left to right):
- 4 × 4 = 16
- 8 ÷ 4 = 2
Addition and Subtraction (left to right):
- 3 + 16 = 19
- 19 - 2 = 17

Result: 17

Method 2: Using Parentheses for Clarity

Expression: 3 + 4 × 2² - 8 ÷ 4

Rewritten with parentheses: 3 + (4 × (2²)) - (8 ÷ 4)

Evaluate each set of parentheses:
- (2²) = 4
- (4 × 4) = 16
- (8 ÷ 4) = 2
Resulting expression: 3 + 16 - 2
Calculate left to right: 3 + 16 = 19, then 19 - 2 = 17

Result: 17

Method 3: Identify and Calculate Terms

Expression: 3 + 4 × 2² - 8 ÷ 4

Identify separate terms: 3, 4 × 2², 8 ÷ 4
Calculate each term:
- 3 remains 3
- 4 × 2² = 4 × 4 = 16
- 8 ÷ 4 = 2
Combine terms with their operations: 3 + 16 - 2 = 17

Result: 17

Interactive Quiz

Test your understanding of the Order of Operations with this interactive quiz:

Key Takeaways

Always follow PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
Multiplication and Division have the same precedence and are evaluated from left to right.
Addition and Subtraction have the same precedence and are evaluated from left to right.
Use parentheses to clarify or change the order of operations if needed.
Be careful with negative numbers and exponents.
Remember that implied multiplication (e.g., 2(3+4)) has the same precedence as explicit multiplication.