Order of Operations: Rules, Examples, and Solved Exercises
Order of Operations: Rules, Examples, and Solved Exercises
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The order of operations is the set of rules that establishes the sequence in which operations within a mathematical expression must be performed. Its importance lies in the fact that the same expression can produce different results if the operations are performed in the wrong order.
When an expression contains only one operation, solving it is usually straightforward. However, when parentheses, exponents, roots, multiplication, division, addition, and subtraction appear together, we need to follow an established procedure. The order of operations ensures that a mathematical expression has a unique result and that different people can solve it by following the same rules.
Key idea: first solve grouping symbols; then exponents and roots; next, multiplication and division; and finally, addition and subtraction. When two operations have the same level of precedence, solve them from left to right.
What Is the Order of Operations?
The order of operations determines the sequence we must follow to correctly evaluate a mathematical expression. It is especially important when the same expression contains two or more types of operations.
For example, consider the expression:
2 + 3 × 4
If we add first, we get:
(2 + 3) × 4 = 5 × 4 = 20
However, if we correctly apply the order of operations, we perform the multiplication first:
2 + 3 × 4 = 2 + 12 = 14
The correct result is 14.
This shows why we cannot arbitrarily choose which operation to perform first. The order of operations establishes a common procedure for interpreting mathematical expressions.
What Is the Order of Operations?
The order of operations can be organized into four main levels:
| Order | Operations | General Rule |
|---|---|---|
| 1 | Parentheses, brackets, and braces | From the inside out |
| 2 | Exponents and roots | Solve before multiplication or division |
| 3 | Multiplication and division | From left to right |
| 4 | Addition and subtraction | From left to right |
A particularly important point is that multiplication and division have the same level of precedence. This does not mean that multiplication must always be performed before division. When both appear in the same expression, they are solved from left to right.
The same applies to addition and subtraction: both have the same level of precedence and are solved from left to right.
1. Grouping Symbols: Parentheses, Brackets, and Braces
Grouping symbols indicate that certain operations must be treated as a unit. The most commonly used symbols are:
- Parentheses: ( )
- Brackets: [ ]
- Braces: { }
When several levels of grouping are present, we generally begin with the operation located deepest inside and work outward.
Example 1: Parentheses
Solve:
8 + (5 − 2)
First, solve what is inside the parentheses:
5 − 2 = 3
Substitute:
8 + 3 = 11
Result: 11.
Example 2: Multiple Grouping Symbols
Solve:
{3 + [5 − (4 − 2)]}
First, solve the innermost parentheses:
4 − 2 = 2
Now:
{3 + [5 − 2]}
Solve the brackets:
5 − 2 = 3
Finally:
3 + 3 = 6
Result: 6.
2. Exponents and Roots in the Order of Operations
After solving the grouping symbols, calculate exponents and roots before moving on to multiplication and division.
Example: Exponent
4 + 3²
First, calculate the exponent:
3² = 9
Then add:
4 + 9 = 13
Result: 13.
Example: Root
7 + √36
First, calculate the root:
√36 = 6
Then:
7 + 6 = 13
Result: 13.
Remember: an exponent or root is not solved after addition or subtraction. In an expression such as 5 + 2³, first calculate 2³ and then add 5.
3. Multiplication and Division: The Same Level of Precedence
One of the most common misconceptions is thinking that multiplication must always be performed before division. This is incorrect: multiplication and division have the same level of precedence.
When they appear together, they must be solved from left to right.
Example
24 ÷ 6 × 2
We begin with the operation that appears first:
24 ÷ 6 = 4
Then:
4 × 2 = 8
Result: 8.
Common mistake: solving 6 × 2 first and then dividing 24 by 12 would produce 2. That procedure changes the order of the expression and does not follow the order of operations.
4. Addition and Subtraction: The Final Level
Addition and subtraction are at the final level of the order of operations. When no higher-priority operations remain, they are performed from left to right.
Example
20 − 7 + 3
First:
20 − 7 = 13
Then:
13 + 3 = 16
Result: 16.
Important: addition and subtraction have the same level of precedence. We should not assume that addition always comes before subtraction.
How to Solve Combined Operations Step by Step
To correctly solve a combined operation, it is helpful to always follow a consistent sequence. This reduces errors and allows us to check each stage of the procedure.
- Identify the grouping symbols.
- Solve the operations inside the grouping symbols.
- Calculate exponents and roots.
- Perform multiplication and division from left to right.
- Perform addition and subtraction from left to right.
- Check the final result.
Complete Example
Solve:
5 + 2 × (7 − 3)²
Step 1: Solve the parentheses.
7 − 3 = 4
The expression becomes:
5 + 2 × 4²
Step 2: Calculate the exponent.
4² = 16
Now:
5 + 2 × 16
Step 3: Perform the multiplication.
2 × 16 = 32
Finally:
5 + 32 = 37
Final result: 37.
Solved Examples of the Order of Operations
Exercise 1
Solve:
8 + 4 × 3
First, multiply:
4 × 3 = 12
Then add:
8 + 12 = 20
Answer: 20.
Exercise 2
Solve:
18 − 12 ÷ 3
First, divide:
12 ÷ 3 = 4
Then:
18 − 4 = 14
Answer: 14.
Exercise 3
Solve:
(6 + 2) × 5
First, solve the parentheses:
6 + 2 = 8
Then:
8 × 5 = 40
Answer: 40.
Exercise 4
Solve:
30 ÷ (2 + 3) × 4
First:
2 + 3 = 5
The expression becomes:
30 ÷ 5 × 4
Division and multiplication have the same level of precedence, so we proceed from left to right:
30 ÷ 5 = 6
6 × 4 = 24
Answer: 24.
Exercise 5
Solve:
10 + 3² × 2
First, calculate the exponent:
3² = 9
Then perform the multiplication:
9 × 2 = 18
Finally:
10 + 18 = 28
Answer: 28.
Exercise 6
Solve:
5 + (3 + 1)²
First, solve the parentheses:
3 + 1 = 4
Then calculate the exponent:
4² = 16
Finally:
5 + 16 = 21
Answer: 21.
Exercise 7
Solve:
5 × 4² − 8 × 2 + 5
First, calculate the exponent:
4² = 16
The expression becomes:
5 × 16 − 8 × 2 + 5
Now perform the multiplications:
80 − 16 + 5
Finally, perform addition and subtraction from left to right:
80 − 16 = 64
64 + 5 = 69
Answer: 69.
Exercise 8
Solve:
12 − 2 × (6 − 3)² ÷ 3
First, solve the parentheses:
6 − 3 = 3
Then calculate the exponent:
3² = 9
The expression becomes:
12 − 2 × 9 ÷ 3
Multiplication and division have the same level of precedence and are solved from left to right:
2 × 9 = 18
18 ÷ 3 = 6
Finally:
12 − 6 = 6
Answer: 6.
Order of Operations with Parentheses, Brackets, and Braces
Expressions can contain several levels of grouping. In these cases, we work from the innermost operation toward the outer grouping symbols.
Advanced Example
Solve:
−6 + {3 − [4 − 2(4 − 7)]}
Step 1: Solve the innermost parentheses.
4 − 7 = −3
The expression becomes:
−6 + {3 − [4 − 2(−3)]}
Step 2: Perform the multiplication.
2(−3) = −6
Therefore:
−6 + {3 − [4 − (−6)]}
Step 3: Solve the brackets.
4 − (−6) = 10
Now:
−6 + {3 − 10}
Step 4: Solve the braces.
3 − 10 = −7
Finally:
−6 + (−7) = −13
Final result: −13.
Common Errors When Applying the Order of Operations
Solving from Left to Right Regardless of the Operation
We should not always begin with the operation that appears furthest to the left. The left-to-right rule applies when the operations have the same level of precedence.
Incorrect: 8 + 2 × 3 → (8 + 2) × 3 = 30.
Correct: 8 + 2 × 3 → 8 + 6 = 14.
Solving the Addition First Because It Appears First
Addition does not have priority over multiplication. If both appear in the same expression and no grouping symbols indicate a different order, multiplication is performed first.
Believing That Multiplication Always Comes Before Division
Multiplication and division belong to the same level of precedence. Therefore, they are solved from left to right.
Ignoring Parentheses
Parentheses can completely change the result of an expression. For example:
2 + 3 × 4 = 14
(2 + 3) × 4 = 20
The only difference is the grouping, but the result changes.
Removing Grouping Symbols Without Considering the Signs
When working with positive and negative numbers, removing parentheses, brackets, or braces requires careful attention to the sign that precedes them. This becomes especially important when working with algebraic expressions.
A Simple Strategy to Avoid Mistakes
A practical way to work with the order of operations is to mentally identify each level before beginning. Instead of trying to solve the entire expression at once, identify which operations have priority.
- Look for grouping symbols: locate parentheses, brackets, and braces.
- Look for exponents and roots: solve them after the grouping symbols.
- Look for multiplication and division: proceed from left to right.
- Finish with addition and subtraction: also proceed from left to right.
This strategy is especially useful when expressions become longer. The goal is not simply to memorize an isolated phrase, but to understand that operations are organized into levels of precedence.
Why Is It Important to Learn the Order of Operations?
The order of operations is fundamental because it provides a precise way to interpret mathematical expressions. Without a common order, the same expression could produce different results depending on the person solving it.
In addition, this knowledge serves as a foundation for later topics such as algebraic expressions, equations, polynomials, functions, and mathematical formulas. Therefore, mastering the order of operations is not limited to solving school exercises: it allows students to correctly interpret more complex mathematical procedures.
It is also important to distinguish between memorizing the order and understanding it. Memorization can be helpful at first, but true mastery develops when students can identify why one operation must be performed before another.
Frequently Asked Questions About the Order of Operations
What Is the Order of Operations?
It is the set of rules that establishes the sequence in which operations within a mathematical expression must be performed in order to obtain a correct and unique result.
What Is Done First in the Order of Operations?
First, solve the grouping symbols, such as parentheses, brackets, and braces, beginning with the innermost operations.
What Is Solved After Parentheses?
Next, solve exponents and roots. Then perform multiplication and division, and finally addition and subtraction.
Which Comes First, Multiplication or Division?
Both have the same level of precedence. When they appear together, they are solved from left to right.
Which Comes First, Addition or Subtraction?
Addition and subtraction have the same level of precedence. When they appear together, they are solved from left to right.
Are Exponents Solved Before Multiplication?
Yes. In an expression without grouping symbols that indicate another order, exponents are calculated before multiplication and division.
Are Roots Solved Before Addition?
Yes. Roots belong to the exponent/root level and are calculated before multiplication, division, addition, and subtraction.
What Does Solving from Left to Right Mean?
It means that when several operations have the same level of precedence, we begin with the one that appears furthest to the left and then continue toward the right.
Why Are Parentheses Important?
Parentheses indicate grouping and can change the order in which operations must be performed. Therefore, an expression with parentheses can have a different result from an apparently similar expression without them.
Conclusion: Understand Before You Calculate
The order of operations establishes a precise sequence for solving mathematical expressions. Its structure can be summarized in four major levels: grouping symbols; exponents and roots; multiplication and division; addition and subtraction.
However, one rule deserves special attention: operations that belong to the same level are solved from left to right. Therefore, multiplication and division do not have different priorities from one another, nor does addition have priority over subtraction.
Learning this procedure helps reduce errors and builds a solid foundation for later topics in arithmetic and algebra. The key is not simply to remember a list, but to learn how to examine an expression, identify its levels of precedence, and solve it step by step.
Rule to remember:
- Grouping symbols.
- Exponents and roots.
- Multiplication and division, from left to right.
- Addition and subtraction, from left to right.
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 Order of Operations.
🎙️ Listen to the full episode here:
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Academic Resource Profile
Collection
Mathematics
Knowledge Center
Arithmetic
Knowledge Area
Order of Operations
Topic
Order of Operations: Rules, Examples, and Solved Exercises
Level
Basic–Intermediate
Estimated Reading Time
12–15 minutes
Resource Type
Master Article
Skills Developed
- Understand the concept of the order of operations.
- Identify the correct sequence for solving mathematical expressions.
- Recognize the function of parentheses, brackets, and braces.
- Solve exponents and roots within combined operations.
- Correctly apply multiplication and division from left to right.
- Solve addition and subtraction while respecting their equal level of precedence.
- Solve combined operations step by step.
- Apply the order of operations with positive and negative numbers.
- Identify common errors when solving mathematical expressions.
- Check results using organized procedures.
- Interpret mathematical expressions accurately.
- Build a foundation for later algebra topics.
The article explicitly establishes the four levels: grouping symbols; exponents and roots; multiplication and division; addition and subtraction, together with the left-to-right rule when operations have the same level of precedence.
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