These worksheets will help reinforce the rules governing the sequence in which mathematical operations should be performed to accurately evaluate expressions. These worksheets typically feature a series of mathematical expressions containing combinations of addition, subtraction, multiplication, division, exponents, and parentheses. By providing exercises that require students to follow the correct order of operations—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right)—these worksheets help learners develop fluency in solving complex mathematical problems while avoiding errors in computation. Through practice and repetition, students enhance their understanding of the hierarchical structure of mathematical operations, enabling them to solve equations and expressions with confidence and accuracy.

**Four Numbers and Three Operations 1** | Answer Key

**Four Numbers and Three Operations 2** | Answer Key

**Five Numbers and Four Operations 1** | Answer Key

**Five Numbers and Four Operations 2** | Answer Key

**Estimating Operations** | **Mental Math Patterns**

**Order of Operations and Equations** | **More Equations**

** Multiplication and Division** | **Place Value / Scientific Notation**

What Are the PEMDAS Order of Operations?

The PEMDAS Order of Operations is a set of rules used to determine the sequence in which mathematical operations should be performed within an expression to obtain the correct result. “PEMDAS” is an acronym that stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). These rules help ensure that mathematical expressions are evaluated consistently and unambiguously, especially when they contain multiple operations.

**Parentheses** – The first step in evaluating an expression according to PEMDAS is to perform any operations enclosed within parentheses or brackets. Parentheses are used to group operations and indicate that the operations within them should be performed first. For example, in the expression 3 x (4 + 2), the operations within the parentheses (4+2) should be evaluated first, resulting in 3 x 6.

**Exponents** – After parentheses, the next step is to evaluate any exponentiation or powers in the expression. Exponents indicate repeated multiplication of a base number by itself a certain number of times. Exponentiation operations are performed from left to right. For example, in the expression 2^{3}4, the exponentiation operation 2^{3} is calculated first, resulting in 8 x 4.

**Multiplication and Division (from left to right) **– Following parentheses and exponents, the next step is to perform multiplication and division operations from left to right. Multiplication and division have the same level of precedence, so they are performed in the order they appear in the expression. For instance, in the expression 6 ÷ 2 x 3, the division 6 ÷ 2 is performed first, followed by the multiplication 3 x 3.

**Addition and Subtraction (from left to right)** – The final step in evaluating an expression according to PEMDAS is to perform addition and subtraction operations from left to right. Addition and subtraction also have the same level of precedence and are performed in the order they appear in the expression. For example, in the expression 5 + 4 – 2, the addition 5 + 4 is performed first, followed by the subtraction 9 – 2.

It’s important to note that PEMDAS establishes a hierarchy of operations, ensuring that calculations are carried out in a systematic and consistent manner. This helps prevent ambiguity and ensures that expressions are evaluated correctly, regardless of their complexity. By following the PEMDAS Order of Operations, mathematicians, scientists, engineers, and students can perform calculations accurately and efficiently, leading to reliable results in various mathematical contexts.

Step-By-Step Examples

**2-Step Problem: 6 + (4 x 3)**

Step 1: Parentheses

6 + (12)

Step 2: Addition

6 + 12 = 18

**4-Step Problem: 8 + (5 x 3) – 2 ^{2}**

Step 1: Parentheses

8 + (15) – 2^{2}

Step 2: Exponents

8 + (15) – 4

Step 3: Addition and Subtraction (Left to Right: Addition In This Case)

23 – 4

Step 4: Subtraction

23 – 4 = 9

**5-Step Problem: 4 + (6 x 2) – 3 ^{2} + 5**

Step 1: Parentheses

4 + (12) – 3^{2} + 5

Step 2: Exponents

4 + (12) – 9 + 5

Step 3: Addition and Subtraction (Left to Right: Addition In This Case)

16 – 9 + 5

Step 4: Addition and Subtraction (Left to Right: Subtraction In This Case)

7 + 5

Step 5: Addition

7 + 5 = 12