PEMDAS Explained: Order of Operations in Math

PEMDAS allows you to solve math problems containing multiple operations. Following the correct order of operations is crucial, otherwise you’ll get the wrong answer!

But what do we mean by multiple operations?

In math, an expression can include different types of calculations—such as addition, subtraction, multiplication, division, and exponents—all in one problem. For example, what is the correct answer to the following expression?

6 + 3 × (2³ − 4) ÷ 2

Without a clear order, solving these expressions can lead to confusion and incorrect answers. That’s where PEMDAS comes in. It’s a simple way to remember the correct order of operations in math.

In this post, you will learn what PEMDAS stands for, how to apply the PEMDAS rule, and why the order of operations matters. We’ll also work through examples, including a complex problem step by step.

What Does PEMDAS Stand For?

PEMDAS is an acronym that represents the order in which to solve parts of a mathematical expression:

• P – Parentheses
• E – Exponents
• MD – Multiplication and Division (left to right)
• AS – Addition and Subtraction (left to right)

Different countries use variations of this acronym. For example, BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, and Subtraction) is common in Canada, while other regions use BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). However, the rules remain the same.

The PEMDAS Rule: Step by Step

The PEMDAS rule ensures that every math problem has only one correct answer. Let’s break it down with examples.

Step 1: Solve Parentheses First

Anything inside parentheses (or brackets) must be solved first.

✅ Correct:
4 × (5 + 3) = 4 × 8 = 32

❌ Wrong:
4 × (5 + 3) = 20 + 3 = 23 (Incorrect because multiplication was done before adding inside the parentheses.)

Step 2: Solve Exponents Next

PEMDAS requires you to simplify any powers or square roots before moving on.

✅ Correct:
5 × 2² = 5 × 4 = 20

❌ Wrong:
5 × 2² = 10² = 100 (Incorrect because exponents must be solved before multiplying.)

Tip: Because a square root is the same as an exponent (power of ½), treat it as an exponent in the order of mathematic operations.

Step 3: Multiplication and Division (Left to Right)

Perform multiplication and division in the order they appear, from left to right.

✅ Correct:
2 + 5 × 3 = 2 + 15 = 17

❌ Wrong:
2 + 5  × 3 = 7 × 3 = 21 (Incorrect because multiplication must be done before addition.)

Step 4: Addition and Subtraction (Left to Right)

Finally, PEMDAS requires you to perform addition and subtraction in order from left to right.

✅ Correct:
30 ÷ 5 × 3 = 6 × 3 = 18

❌ Wrong:
30 ÷ 5 × 3 = 30 ÷ 15 = 2 (Incorrect because division and multiplication must be solved in order from left to right.)

Common Mistakes When Using PEMDAS

1. Ignoring Parentheses: Always solve inside parentheses first, even if it changes the usual order.
   • ❌ Wrong: 4 + 3 × (2 + 1) = 4 + 3 × 3 = 7 × 3 = 21
   • ✅ Correct: 4 + 3 × (2 + 1) = 4 + 9 = 13
2. Mixing Up Multiplication and Addition: Multiplication comes before addition, even if addition appears first in the expression.
   • ❌ Wrong: 2 + 5 × 3 = 7 × 3 = 21
   • ✅ Correct: 2 + 5 × 3 = 2 + 15 = 17
3. Forgetting Left-to-Right Rule: Multiplication and division, as well as addition and subtraction, must be performed in order from left to right.
   • ❌ Wrong: 30 ÷ 5 × 3 = 30 ÷ 15 = 2
   • ✅ Correct: 30 ÷ 5 × 3 = 6 × 3 = 18
4. Confusion with Multiple Parentheses: If there are nested parentheses, always start with the innermost set first.
   • ❌ Wrong: 6 + [(4 + 2) × 3] ÷ 2 = 6 + [4 + 6] ÷ 2 = 6 + 10 ÷ 2 = 6 + 5 = 11
   • ✅ Correct: 6+ [(4 + 2) × 3] ÷ 2 = 6 + [6 × 3] ÷ 2 = 6 + 18 ÷ 2 = 6 + 9 = 15
5. Exponents with Exponents: When you have an exponent raised to another exponent, solve from the top down.
   • ❌ Wrong: 4^3^2 = (4³)² = 64² = 4096
   • ✅ Correct: 4^3^2 = 4⁹ = 262144

A Complex PEMDAS Example: Step-by-Step Breakdown

Let’s apply PEMDAS to a more complicated expression:

6 + 3 × (2³ − 4) ÷ 2

Step 1: Parentheses

Solve inside the parentheses first:
2³ – 4 = 8 – 4 = 4

So the expression simplifies to:
6 + 3 × 4 ÷ 2

Step 2: Exponents

Already solved inside parentheses.

Step 3: Multiplication and Division (Left to Right)

3 × 4 = 12
12 ÷ 2 = 6

Now we have:
6 + 6

Step 4: Addition

6 + 6 = 12

✅ Final Answer: 12

PEMDAS Summary

Following PEMDAS ensures that everyone gets the same correct answer when solving math problems. Remember: Parentheses first, then Exponents, followed by Multiplication and Division (left to right), and finally, Addition and Subtraction (left to right). Mastering these rules makes complex math easier and helps avoid common mistakes.

In this post, you learned what PEMDAS stands for, how to apply the PEMDAS rule, and why the order of operations matters. Now, try solving a few expressions on your own to test your understanding!

You can use PEMDAS to help you find the correct solutions when using statistical formulas for things such as calculating correlation coefficients, standard deviations, standard error of the mean, and the values for a linear regression line.