How to Evaluate Order Of Operations Step by Step with Rules and Solved Examples
The Evaluating Order Of Operations is a fundamental concept in maths that makes sure mathematical expressions with more than one operation are solved correctly and consistently. Whether you’re preparing for exams, doing homework, or simply solving daily math problems, understanding the rules of order of operations is essential. At Vedantu, we break down complex maths topics into simple, practical steps so every student can succeed.
What is the Order of Operations?
The order of operations refers to the sequence in which operations like brackets/parentheses, exponents, multiplication, division, addition, and subtraction should be performed in a mathematical expression. If these steps are not followed, you may end up with incorrect answers. To standardize this process worldwide, acronyms like PEMDAS or BODMAS are used. These are critical for all school and competitive exams, and they help avoid confusion when solving arithmetic or algebraic expressions.
- PEMDAS stands for: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
- BODMAS stands for: Brackets, Orders, Division, Multiplication, Addition, Subtraction.
For example, in the question 8 + 12 × 2: If you solve left to right, you might calculate (8 + 12) × 2 = 20 × 2 = 40. But with the correct order, you do 12 × 2 = 24 first, then 8 + 24 = 32.
Order of Operations Rules: PEMDAS and BODMAS
These acronyms help you remember the priority for each operation:
| PEMDAS | BODMAS |
|---|---|
|
|
Multiplication and Division are of equal priority; do them from left to right. The same applies to Addition and Subtraction.
Step-by-Step Procedure for Evaluating Expressions
- Solve expressions inside Brackets/Parentheses first.
- Calculate Exponents or Orders next (like squares or square roots).
- Do all Multiplication and Division, proceeding left to right.
- Finish with Addition and Subtraction, again from left to right.
This approach ensures every part of a complex arithmetic or algebraic expression is handled in the right sequence.
Worked Examples
Let’s apply the rules through some examples:
-
Example 1: Solve 6 + 2 × 3
- Step 1: Multiplication first (2 × 3 = 6)
- Step 2: Addition 6 + 6 = 12
Final answer: 12
-
Example 2: Solve 8 + [4 × (6 - 2)²]
- Step 1: Parentheses: (6 - 2) = 4
- Step 2: Exponent: 4² = 16
- Step 3: Multiplication: 4 × 16 = 64
- Step 4: Addition: 8 + 64 = 72
Final answer: 72
-
Example 3: Evaluate 24 ÷ 4 × 2
- Left to right:
- Step 1: Division 24 ÷ 4 = 6
- Step 2: Multiplication 6 × 2 = 12
Final answer: 12
-
Example 4: Solve [2 + 3 × (4² - 1)]
- Step 1: Inside parentheses: 4² = 16; 16 - 1 = 15
- Step 2: Multiplication: 3 × 15 = 45
- Step 3: Addition: 2 + 45 = 47
Final answer: 47
Practice Problems
- Simplify 5 + 18 ÷ 3 × 2
- Evaluate (7 + 3) × 2²
- Solve 12 - 2 × (8 - 3)
- Find the value of 6 × [2 + (3 × 2)]
- Calculate 36 ÷ 3 - 2²
Try solving these using PEMDAS/BODMAS rules. (Check your answers at the end for self-assessment!)
Common Mistakes to Avoid
- Doing addition or subtraction before multiplication/division.
- Forgetting to process brackets/parentheses first.
- Not following left-to-right for multiplication/division or addition/subtraction when both are present.
- Ignoring exponents (orders) or solving them after multiplication.
- Rushing through without double-checking each step.
Real-World Applications
Order of operations appears everywhere in daily life and advanced fields. For instance, when using a calculator, entering the wrong sequence gives the wrong total; in coding and computer algorithms, correct mathematical precedence avoids costly errors. Even splitting bills or calculating discounts uses these rules. At Vedantu, we show how mastering these basics supports not just maths exams but also smart, error-free living!
In this topic, you learned how crucial the Evaluating Order Of Operations is for solving mathematical expressions accurately. By following PEMDAS/BODMAS, you can avoid mistakes, speed up problem-solving, and strengthen your foundation for both school assessments and real-life applications. Keep practicing these steps for more confidence in maths. For more on this topic, check out our Order of Operations and BODMAS Rules articles, or explore topics like Evaluating Expressions and Laws of Exponents at Vedantu.
FAQs on Evaluating Order Of Operations in Math Expressions
1. What is the order of operations in math?
The order of operations is the rule that tells you the sequence to follow when solving mathematical expressions: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. This is often remembered as PEMDAS or BODMAS.
- P – Parentheses (or Brackets)
- E – Exponents (or Orders)
- M/D – Multiplication and Division (from left to right)
- A/S – Addition and Subtraction (from left to right)
2. Why is the order of operations important?
The order of operations is important because it guarantees a single correct answer for any mathematical expression. Without it, expressions like 3 + 4 × 2 could give different results.
- Correct method: 3 + (4 × 2) = 3 + 8 = 11
- Incorrect method: (3 + 4) × 2 = 7 × 2 = 14
3. How do you apply PEMDAS step by step?
To apply PEMDAS, solve parts of the expression in the correct priority order from left to right. For example, evaluate 6 + 2 × (3² − 1):
- Parentheses: 3² − 1
- Exponents: 3² = 9
- Parentheses: 9 − 1 = 8
- Multiplication: 2 × 8 = 16
- Addition: 6 + 16 = 22
4. Do you multiply or divide first in the order of operations?
You perform multiplication and division from left to right, whichever comes first in the expression. They have equal priority in the order of operations.
- Example: 20 ÷ 4 × 2
- Step 1: 20 ÷ 4 = 5
- Step 2: 5 × 2 = 10
5. Do you add or subtract first in PEMDAS?
You perform addition and subtraction from left to right, depending on which appears first. They have the same level of priority.
- Example: 15 − 5 + 3
- Step 1: 15 − 5 = 10
- Step 2: 10 + 3 = 13
6. What comes first, parentheses or exponents?
In the order of operations, parentheses come before exponents. You must simplify everything inside parentheses before applying powers.
- Example: (2 + 3)²
- Step 1: Parentheses: 2 + 3 = 5
- Step 2: Exponent: 5² = 25
7. Can you give an example of order of operations with exponents?
An example of order of operations with exponents is evaluating 4 + 2³ × 3. Follow PEMDAS:
- Exponents: 2³ = 8
- Multiplication: 8 × 3 = 24
- Addition: 4 + 24 = 28
8. What is the difference between PEMDAS and BODMAS?
The difference between PEMDAS and BODMAS is only in wording, not in mathematical meaning. Both describe the same order of operations.
- PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
- BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction
9. How do you solve order of operations with multiple parentheses?
To solve expressions with multiple parentheses, simplify the innermost parentheses first and work outward. For example: 5 × (2 + (6 − 3)):
- Inner parentheses: 6 − 3 = 3
- Next parentheses: 2 + 3 = 5
- Multiplication: 5 × 5 = 25
10. What are common mistakes when evaluating order of operations?
Common mistakes in evaluating order of operations include ignoring left-to-right rules and skipping parentheses. The most frequent errors are:
- Doing addition before multiplication (e.g., 2 + 3 × 4)
- Multiplying before dividing regardless of position
- Forgetting to simplify inside parentheses first
- Ignoring exponents
