Combined Operations: How to Solve Them Step by Step with Examples
Combined Operations: How to Solve Them Step by Step with Examples
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Combined operations are mathematical expressions that contain two or more different operations, such as addition, subtraction, multiplication, division, exponents, or roots. To solve them correctly, it is not enough to calculate from left to right without considering the operations involved; we must follow an established order.
Learning how to solve combined operations allows us to correctly interpret numerical expressions and provides an important foundation for advancing to topics such as algebraic expressions, equations, polynomials, and functions. In this article, you will learn what combined operations are, how to solve them step by step, how to work with grouping symbols and exponents, which mistakes are most common, and how to check your results.
Key idea: to solve combined operations, we must follow the order of operations. First, solve grouping symbols; next, exponents and roots; then, multiplication and division; and finally, addition and subtraction. Operations that have the same level of precedence are solved from left to right.
What Are Combined Operations?
A combined operation is a mathematical expression that contains different operations within the same expression. For example:
8 + 3 × 4
This expression contains addition and multiplication. We cannot perform both at the same time or arbitrarily choose which one to do first. We must apply the order of operations.
Multiplication has a higher level of precedence than addition, so we first calculate:
3 × 4 = 12
Then we perform the addition:
8 + 12 = 20
Result: 20.
The difference between a simple operation and a combined operation therefore lies in the need to organize the procedure. The more operations an expression contains, the more important it becomes to correctly identify the order in which they must be performed.
What Is the Order for Solving Combined Operations?
To correctly solve an expression, we must apply the levels of the order of operations. The general procedure is as follows:
| Order | What to Solve |
|---|---|
| 1 | Grouping symbols |
| 2 | Exponents and roots |
| 3 | Multiplication and division |
| 4 | Addition and subtraction |
There is one fundamental point to remember: multiplication and division have the same level of precedence. Therefore, when they appear together, they are solved from left to right.
The same applies to addition and subtraction. Neither has priority over the other; when they occur at the same level, they are solved from left to right.
Practical rule: this does not simply mean "do multiplication first." We must identify the level of each operation and follow the correct order within each level.
How to Solve Combined Operations Step by Step
An effective way to solve combined operations is to work in stages. Instead of trying to calculate the entire expression mentally, simplify one part at a time.
- Identify the parentheses, brackets, or braces.
- Solve the operations inside the grouping symbols.
- Calculate the exponents and roots.
- Perform multiplication and division from left to right.
- Perform addition and subtraction from left to right.
- Check the result.
Basic Example
Solve:
7 + 5 × 2
We do not begin with the addition because multiplication has a higher level of precedence.
Step 1: Multiply:
5 × 2 = 10
Step 2: Add:
7 + 10 = 17
Result: 17.
Combined Operations with Parentheses
Parentheses are grouping symbols. Their presence indicates that certain operations must be considered before the operations outside them.
Observe the difference between these two expressions:
5 + 3 × 2
(5 + 3) × 2
In the first expression, we perform the multiplication first:
5 + 6 = 11
In the second, the parentheses require us to solve the addition first:
(5 + 3) × 2 = 8 × 2 = 16
The results are different because the parentheses change the order of the operations.
Example with Parentheses
Solve:
18 − (4 + 2) × 2
Step 1: Solve the parentheses:
4 + 2 = 6
The expression becomes:
18 − 6 × 2
Step 2: Perform the multiplication:
6 × 2 = 12
Step 3: Perform the subtraction:
18 − 12 = 6
Result: 6.
Combined Operations with Exponents and Roots
Exponents and roots can also appear within combined operations. These operations are solved before multiplication and division.
Example with an Exponent
Solve:
6 + 2³ × 2
Step 1: Calculate the exponent:
2³ = 8
The expression becomes:
6 + 8 × 2
Step 2: Multiply:
8 × 2 = 16
Step 3: Add:
6 + 16 = 22
Result: 22.
Example with a Root
Solve:
4 + √25 × 2
Step 1: Calculate the root:
√25 = 5
Now we have:
4 + 5 × 2
Step 2: Multiply:
5 × 2 = 10
Step 3: Add:
4 + 10 = 14
Result: 14.
Combined Operations with Multiplication and Division
When an expression contains multiplication and division, both operations have the same level of precedence. Therefore, we must proceed from left to right.
Example
Solve:
36 ÷ 6 × 3
First, perform the operation that appears furthest to the left:
36 ÷ 6 = 6
Then:
6 × 3 = 18
Result: 18.
Common mistake: we should not assume that multiplication is always performed before division. Both operations have the same level of precedence.
Another Example
Solve:
48 ÷ 4 × 2 ÷ 3
Proceed from left to right:
48 ÷ 4 = 12
12 × 2 = 24
24 ÷ 3 = 8
Result: 8.
Combined Operations with Addition and Subtraction
Addition and subtraction also have the same level of precedence. When no higher-priority operations remain, they are solved from left to right.
Solve:
25 − 8 + 4 − 3
We begin with the operation furthest to the left:
25 − 8 = 17
Then:
17 + 4 = 21
Finally:
21 − 3 = 18
Result: 18.
Solved Combined Operations Exercises
The following exercises allow you to progressively apply the rules explained above. The difficulty increases as more operations and grouping symbols are introduced.
Exercise 1
9 + 6 × 2
First, multiply:
6 × 2 = 12
Then:
9 + 12 = 21
Answer: 21.
Exercise 2
30 − 18 ÷ 3
First, divide:
18 ÷ 3 = 6
Then:
30 − 6 = 24
Answer: 24.
Exercise 3
(8 + 4) ÷ 3
First, solve the parentheses:
8 + 4 = 12
Then:
12 ÷ 3 = 4
Answer: 4.
Exercise 4
7 × (5 − 2) + 4
First:
5 − 2 = 3
Then multiply:
7 × 3 = 21
Finally:
21 + 4 = 25
Answer: 25.
Exercise 5
20 − 2³ + 6
First, calculate the exponent:
2³ = 8
The expression becomes:
20 − 8 + 6
Now proceed from left to right:
20 − 8 = 12
12 + 6 = 18
Answer: 18.
Exercise 6
5 + 3 × (10 − 6)
Parentheses:
10 − 6 = 4
Multiplication:
3 × 4 = 12
Addition:
5 + 12 = 17
Answer: 17.
Exercise 7
40 ÷ 5 + 3 × 2
First, perform the division and multiplication:
40 ÷ 5 = 8
3 × 2 = 6
Finally:
8 + 6 = 14
Answer: 14.
Exercise 8
(6 + 2)² ÷ 4
First, solve the parentheses:
6 + 2 = 8
Then calculate the exponent:
8² = 64
Finally:
64 ÷ 4 = 16
Answer: 16.
Continue Learning with Frecuencia Educativa
This topic is part of the episode:
T2E4 | Arithmetic Fundamentals: Exponents, Roots, Operations, and Absolute Value | Frecuencia Educativa
In this episode, you will find an integrated explanation of four fundamental topics in Arithmetic and the relationships between them.
In this article, we take a closer look at Combined Operations.
🎙️ Listen to the full episode here:
Module Infographic
You can also check out the infographic for a visual summary of the main concepts covered in this topic.
You can review the key ideas, concepts, and procedures presented in this module.
Academic Resource Profile
Collection
Mathematics
Knowledge Center
Arithmetic
Knowledge Area
Combined Operations
Topic
Combined Operations: How to Solve Them Step by Step with Examples
Level
Basic–Intermediate
Estimated Reading Time
12–15 minutes
Resource Type
Master Article
Skills Developed
- Understand what combined operations are.
- Identify the different operations that may appear in a numerical expression.
- Correctly apply the order of operations.
- Solve expressions with grouping symbols.
- Solve combined operations involving exponents and roots.
- Solve expressions containing multiplication and division.
- Correctly apply the left-to-right rule to operations with the same level of precedence.
- Solve expressions containing addition and subtraction.
- Solve combined operations step by step.
- Identify common procedural errors.
- Check the results obtained.
- Use combined operations as a foundation for later algebra topics.
The article progressively develops basic operations, parentheses, exponents, roots, multiplication and division, addition and subtraction, as well as solved exercises.
Library Path
Digital Library → Mathematics Collection → Knowledge Center: Arithmetic → Knowledge Area: Combined Operations
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