Evaluating algebraic expressions doesn’t have to be overwhelming — if you know the right steps.
At the heart of solving equations lies the P.E.M.D.A.S. rule, a simple yet powerful guideline that ensures your calculations are accurate.
This rule helps you prioritize operations like parentheses, exponents, multiplication, and more, preventing common mistakes that can derail your work.
Think of P.E.M.D.A.S. as a roadmap for navigating the complexities of math.
When followed correctly, it transforms even the trickiest equations into manageable tasks.
In this guide, you’ll discover everything you need to master this essential tool and apply it confidently in any situation.
What is P.E.M.D.A.S.?
P.E.M.D.A.S. is an acronym for the sequence of operations you must follow when solving mathematical equations:
- Parentheses: Solve expressions inside parentheses or brackets first.
- Exponents: Simplify powers and roots next.
- Multiplication and Division: Perform these from left to right.
- Addition and Subtraction: Complete these last, also working from left to right.
To make it easier to remember, many students use the phrase:
“Please Excuse My Dear Aunt Sally.”
For example, in the equation 5 + 3 × 2, the proper order is to handle multiplication before addition:
- First, multiply: 3 × 2 = 6
- Then, add: 5 + 6 = 11
If you approached this equation left to right without following P.E.M.D.A.S., you might incorrectly add first (5 + 3 = 8) and then multiply, resulting in 16. This highlights the importance of following the order.
Why is P.E.M.D.A.S. important?
Without a standard order of operations, solving equations would become chaotic. That’s because different people would arrive at different answers for the same problem.
P.E.M.D.A.S. establishes a universal framework for approaching calculations.
Consider the equation 8 ÷ 4 × 2:
- Following the left-to-right rule for multiplication and division, the correct answer is 4 × 2 = 8.
- If you perform these out of sequence, such as dividing the product of 4 × 2 by 8, you would get 0.5—an entirely different answer.
The order of operations ensures clarity, precision, and consistency in mathematics, whether you’re working on a simple calculation or a complex algebraic expression.
The step-by-step process of P.E.M.D.A.S.
1. Parentheses: organizing the problem
Parentheses help group parts of an equation that must be solved first.
They simplify calculations by prioritizing specific operations, often making the problem more manageable.
For example, consider:
2 × (3 + 5)
Here, you solve the parentheses first:
3 + 5 = 8
Then multiply:
2 × 8 = 16
When equations have multiple sets of parentheses, work from the innermost group outward. For example:
3 × (2 + (1 + 4))
- Solve the inner parentheses: 1 + 4 = 5
- Then the outer parentheses: 2 + 5 = 7
- Finally, multiply: 3 × 7 = 21
2. Exponents: simplifying powers and roots
Exponents represent repeated multiplication, such as 3² = 3 × 3 = 9. Always handle exponents after parentheses but before other operations.
For example:
(2 + 3)²
- Solve the parentheses first: 2 + 3 = 5
- Then apply the exponent: 5² = 25
Roots, such as square roots (√), also fall under this step. For example:
√(16 + 9)
- Solve inside the parentheses: 16 + 9 = 25
- Then find the square root: √25 = 5
3. Multiplication and division: working left to right
Multiplication and division have equal priority, so they are performed in the order they appear from left to right.
For example:
12 ÷ 3 × 2
- Start with division: 12 ÷ 3 = 4
- Then multiply: 4 × 2 = 8
If the order is reversed (e.g., multiplying before dividing), you’ll arrive at the wrong answer. Always remember to move left to right for these operations.
4. Addition and subtraction: finishing the equation
Finally, perform addition and subtraction, which also have equal priority. Like multiplication and division, work from left to right.
For example:
20 – 5 + 3
- First subtract: 20 – 5 = 15
- Then add: 15 + 3 = 18
When combined with parentheses or other operations, addition and subtraction are always the last steps.
A real-world example of P.E.M.D.A.S. in action
Let’s solve the following equation using P.E.M.D.A.S.:
4 × (6 + 2²) ÷ 2 – 5
- Parentheses: Solve inside the parentheses:
6 + 2² - Exponents: Simplify the exponent first:
2² = 4
Now the parentheses are: 6 + 4 = 10
Update the equation: 4 × 10 ÷ 2 – 5 - Multiplication and division: Work left to right:
4 × 10 = 40
40 ÷ 2 = 20
The equation is now: 20 – 5 - Addition and subtraction: Complete the final step:
20 – 5 = 15
The final answer is 15.
Common mistakes and how to avoid them
1) Overlooking parentheses
Parentheses are a cornerstone of the order of operations, as they dictate which parts of an equation take precedence.
Ignoring parentheses or misinterpreting nested ones can lead to entirely incorrect answers.
For example, in the equation 3 × (4 + 5) − 2, solving it without prioritizing the parentheses would yield a wildly different result.
Nested parentheses, where one set is enclosed within another (e.g., (2 × (3 + 4)) − 5), can be particularly tricky. The key is to always start with the innermost set of parentheses and work outward.
To avoid mistakes:
- Take your time to identify all parentheses in the equation.
- Write each step clearly as you solve.
- Double-check that every operation within parentheses is completed before moving on.
Skipping this essential step will lead to errors that could snowball as the problem progresses.
2) Forgetting the left-to-right rule
The left-to-right rule is critical for handling operations of equal priority, such as multiplication and division or addition and subtraction.
Many people mistakenly apply multiplication before division or addition before subtraction, regardless of their order in the equation.
For instance, in the equation 12 ÷ 3 × 4, performing multiplication first (3 × 4 = 12, then 12 ÷ 12 = 1) gives an incorrect result.
Following the left-to-right rule, you divide first: 12 ÷ 3 = 4, then multiply: 4 × 4 = 16.
To avoid this mistake:
- Move systematically through the equation from left to right after handling parentheses and exponents.
- Don’t assume one operation takes priority over another unless P.E.M.D.A.S. specifies it.
- Practice equations that test your understanding of this rule to build confidence.
Overlooking this principle often leads to answers that feel logical but are mathematically incorrect.
3) Rushing through exponents
Exponents represent powers and are crucial in determining the correct value of an expression.
Rushing through this step can lead to mistakes, especially when dealing with fractional exponents, negative bases, or complex equations involving multiple powers.
Consider the equation (2 + 3)² − 4. Solving it hastily might lead someone to square only the 3 instead of the entire result of (2 + 3), giving 9 − 4 = 5 instead of the correct answer, 25 − 4 = 21.
To avoid this error:
- Carefully identify all exponents in the equation before proceeding.
- Double-check calculations for fractional or negative exponents, as these often trip people up.
- Use parentheses when writing the equation to clarify what the exponent applies to, such as (−3)² versus −3², which yield different results.
Taking the time to handle exponents accurately ensures your calculations remain consistent throughout the problem.
Practice problems to sharpen your skills
Try solving these equations step by step using P.E.M.D.A.S.:
- 8 + 3 × (2² + 1)
- (5 × 4) ÷ 2 + 6
- 12 ÷ (3 + 3) × 2² – 1
Check your work to ensure you’re following the order correctly.
The more you practice, the easier these steps will become.
Final thoughts
Math doesn’t have to be a guessing game.
With P.E.M.D.A.S. in your toolkit, you can approach algebraic expressions with clarity and precision.
In order to avoid common errors and build a solid foundation for tackling more advanced problems, you just need to do one thing:
Understand how the order of operations works and practice these techniques.
So, the next time you face a challenging equation, remember: solving it is simply a matter of taking it step by step.
Mastering P.E.M.D.A.S. isn’t just about getting the right answer — it aims to help you build the confidence to handle any math problem that comes your way.