Why P.E.M.D.A.S. matters: The secret to solving algebra problems

When solving algebraic expressions, one of the biggest challenges is ensuring that each operation is performed in the correct order.

Without the proper sequence, your results may be inaccurate, even if you understand the basic arithmetic involved.

This is where the order of operations, often memorized through the acronym P.E.M.D.A.S. (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction), becomes crucial.

In this guide, we will walk through the process of evaluating algebraic expressions by applying the order of operations, complete with examples and tables for clarity.

Table of Contents

Why is the order of operations important?

Imagine receiving the following problem:

3 + 5 × 2

Without a clear understanding of P.E.M.D.A.S., some may tackle this problem by adding 3 and 5 first, giving 8, and then multiplying by 2, resulting in 16.

However, the correct approach follows the order of operations.

The basic rule is that multiplication comes before addition. Thus, you first calculate 5 × 2 = 1, and then add 3. That’s how it results in 13.

As you can see, the order in which you solve these problems dramatically changes the result.

Let’s dive deeper into P.E.M.D.A.S. to help you solve more complex expressions correctly.

Breaking Down P.E.M.D.A.S.

Parentheses (P): Begin by solving any expressions inside parentheses.
Exponents (E): Next, tackle any exponents (powers or square roots).
Multiplication and Division (MD): Perform multiplication and division as they appear from left to right. These operations are of equal importance.
Addition and Subtraction (AS): Finally, carry out addition and subtraction, also from left to right.

To help you remember this sequence, use the mnemonic: “Please Excuse My Dear Aunt Sally.”

Breaking down P.E.M.D.A.S.

Understanding and mastering P.E.M.D.A.S. is essential for evaluating algebraic expressions accurately.

It represents the sequence in which operations must be performed:

Parentheses (P): Begin by solving any expressions inside parentheses.
Exponents (E): Next, tackle any exponents (powers or square roots).
Multiplication and Division (MD): Perform multiplication and division as they appear from left to right. These operations are of equal importance.
Addition and Subtraction (AS): Finally, carry out addition and subtraction, also from left to right.

To help you remember this sequence, use the mnemonic: “Please Excuse My Dear Aunt Sally.”

Below, we will break down each component to better understand how it functions in solving algebraic expressions:

1) Parentheses

Parentheses are the first operation to be addressed in any mathematical expression. Anything inside parentheses is solved first to simplify the rest of the problem.

Parentheses can group numbers or variables together, often to clarify the order in which operations are performed.

For example, in the expression $\times (4 + 5)$ , the addition inside the parentheses must be solved first.

Solving $4 + 5 = 9$ gives us $\times 9 = 27$ . Without addressing the parentheses first, you could end up solving the equation incorrectly.

Parentheses are also useful when breaking down more complex expressions into smaller, manageable parts.

2) Exponents

Exponents represent repeated multiplication and come second in the P.E.M.D.A.S. hierarchy.

This step simplifies expressions where a number is raised to a power.

For instance, $2^3$ (read as “two to the power of three”) equals $\times 2 \times 2 = 8$ .

Exponents can often make equations seem more complicated, but by isolating and solving them right after parentheses, you can greatly simplify the expression.

For example, in $2^2 \times 3$ , first solve the exponent $2^2 = 4$ , then continue with multiplication and addition.

In general, exponents help you handle not only algebraic expressions but also more advanced mathematical concepts.

3) Multiplication and Division (from Left to Right)

Multiplication and division are of equal precedence, meaning you perform them from left to right in the order they appear. It is a common mistake to assume multiplication always comes before division because of the sequence in the acronym.

For example, in the expression $20/4 \times 5$ , you first divide $20/4 = 5$ , and then multiply $\times 5 = 25$ .

Being aware of this rule ensures that you maintain accuracy, especially in more complicated equations.

In algebra, you’ll often encounter variables being multiplied by numbers, making it crucial to follow this rule precisely.

4) Addition and Subtraction (from Left to Right)

Just like multiplication and division, addition and subtraction share the same level of importance. You should perform these operations from left to right in the order they appear.

For instance, in the equation $10 - 5 + 3$ , you subtract first: $10 - 5 = 5$ , and then add $5 + 3 = 8$ .

In algebraic expressions, addition and subtraction often combine terms, especially when dealing with variables.

For example, when simplifying $x + 5 - 3$ , you first subtract 3 from 5 before adding it to the variable term.

Performing these operations in the correct order is key to simplifying expressions accurately.

Step-by-Step example using P.E.M.D.A.S.

Let’s work through the expression:

5 * 9 – (2 + 4)² / 2 + (15 – 5)

Parentheses: Solve inside parentheses first.
- $2 + 4 = 6$
- $15 - 5 = 10$
Expression becomes:
- $5 * 9 - (6)^{2} /2 + 10$
Exponents: Now solve the exponent.
- $6^{2} = 36$
Expression becomes:
- $5 * 9 - 36/2 + 10$
Multiplication and Division (left to right):
- $5 * 9 = 45$
- $36/2 = 18$
Expression becomes:
- $45 - 18 + 10$
Addition and Subtraction (left to right):
- $45 - 18 = 27$
- $27 + 10 = 37$

Thus, the correct answer is 37.

A second example to reinforce the concept

Let’s evaluate the expression:

3 * (5 + 8) – 2² / 4 + 3

Parentheses: First, solve what’s inside the parentheses.
- $5 + 8 = 13$
Expression becomes:
- $3 * 13 - 2^{2} /4 + 3$
Exponents: Solve any exponents.
- $2^{2} = 4$
Expression becomes:
- $3 * 13 - 4/4 + 3$
Multiplication and Division (left to right):
- $3 * 13 = 39$
- $4/4 = 1$
Expression becomes:
- $39 - 1 + 3$
Addition and Subtraction (left to right):
- $39 - 1 = 38$
- $38 + 3 = 41$

Thus, the correct answer is 41.

Practice table

Use the following table to practice evaluating algebraic expressions using P.E.M.D.A.S.

Expression	Parentheses	Exponents	Multiplication/Division	Addition/Subtraction	Result
$4 * (6 + 3) - 5$	$6 + 3 = 9$	None	$4 * 9 = 36$	$36 - 5 = 31$	31
$8 + (3^{2}) /9 - 1$	None	$3^{2} = 9$	$9/9 = 1$	$8 + 1 - 1 = 8$	8
$(10 - 6)^{2} /2 + 4$	$10 - 6 = 4$	$4^{2} = 16$	$16/2 = 8$	$8 + 4 = 12$	12
$3 * (7 + 2) - 2^{2} + (5 - 3)$	$7 + 2 = 9$ , $5 - 3 = 2$	$2^{2} = 4$	$3 * 9 = 27$	$27 - 4 + 2 = 25$	25

Common mistakes to avoid with P.E.M.D.A.S.

Even with the P.E.M.D.A.S. rule, it’s easy to make mistakes.

Here are a few tips to avoid common pitfalls:

Skipping steps: Often, people skip steps when they feel the operations are simple, which can lead to errors. Always follow the correct sequence, even for simple operations. Jumping ahead can lead to errors.
Assuming the order of the acronym: Many believe multiplication always comes before division, or that addition comes before subtraction. Remember to evaluate them from left to right.
Neglecting parentheses: Failing to address parentheses first can throw off your entire solution, particularly with more complex algebraic expressions.

Wrapping up

Mastering the order of operations is crucial for solving algebraic expressions correctly. By following the P.E.M.D.A.S. rule, you can systematically break down even the most complex problems.

Practice with different expressions, and over time, the process will become second nature.

Whether you’re working through basic algebra problems or tackling more advanced equations, using the correct order of operations ensures accuracy every time.

So remember, Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction—in that order!

With consistent practice, you’ll soon find that evaluating algebraic expressions isn’t as daunting as it first appears.