Order of Operations: The Rules of Mathematical Calculation

The PEMDAS Breakdown

• Parentheses: Perform all operations inside grouping symbols first.
• Exponents: Calculate powers and roots (e.g., $x^2$ or $\sqrt{x}$).
• Multiplication and Division: These hold equal priority.
• Addition and Subtraction: These also hold equal priority.

The BODMAS Breakdown

• Brackets: Equivalent to Parentheses.
• Orders: Equivalent to Exponents (powers and roots).
• Division and Multiplication: Equal priority.
• Addition and Subtraction: Equal priority.

Navigating the "Equal Priority" Trap

The most common mistake students make is assuming that Multiplication always comes before Division, or that Addition always comes before Subtraction. This is incorrect. Multiplication and Division are "partners"—they exist on the same level of the hierarchy. The same applies to Addition and Subtraction.

When you encounter both in an expression, you must solve them from left to right.

Worked Example 1: The Left-to-Right Rule

Consider the expression: $12 \div 3 \cdot 2$

If you mistakenly thought Multiplication always comes before Division, you might calculate: $12 \div (3 \cdot 2) = 12 \div 6 = 2 \quad \text{(Incorrect!)}$

The correct way is to follow the left-to-right rule for equal priority:

$$\begin{aligned} 12 \div 3 &= 4 \\ 4 \cdot 2 &= 8 \end{aligned}$$

The correct answer is $8$.

Step-by-Step Mastery

To solve complex expressions without errors, it is best to treat the process like a checklist. Let's look at a multi-step problem:

Problem: $5 + (6 - 2)^2 \cdot 3$

Step 1: Parentheses (Grouping) First, look inside the parentheses. $6 - 2 = 4$ The expression becomes: $5 + 4^2 \cdot 3$

Step 2: Exponents Next, handle the power. $4^2 = 16$ The expression becomes: $5 + 16 \cdot 3$

Step 3: Multiplication and Division Now, perform the multiplication before the addition. $16 \cdot 3 = 48$ The expression becomes: $5 + 48$

Step 4: Addition and Subtraction Finally, complete the addition. $5 + 48 = 53$

Exam Tip: When writing out your work on a test, try to rewrite the entire expression after every single step. This "vertical" method makes it much easier for you (and your teacher) to spot exactly where a calculation error occurred.

Summary Table of Operations

By mastering these rules, you ensure that your mathematical logic is sound and that your results are consistent with the rest of the scientific community.