Apply Order Of Operations — Evaluate expressions using order of operations.

Historical Context & Motivation

Before mathematicians agreed on a standard set of rules for evaluating expressions, the same string of numbers and operations could produce wildly different results depending on who was reading it. Consider a simple expression like 3 + 4 × 2: one person might calculate 14, while another might get 11. In the skilled trades, where a miscalculation can mean a blown circuit or an undersized conductor, this kind of ambiguity is unacceptable. The order of operations emerged over centuries as a shared convention that eliminates confusion and guarantees that every person evaluating the same expression arrives at the same answer. Understanding how this convention developed helps explain why each rule exists and why the hierarchy is structured the way it is.

The central question that order of operations answers is straightforward: when an expression contains multiple operations, which do you perform first? Without a universal convention, formulas for voltage drops, conduit fill calculations, and amperage ratings would be dangerously ambiguous. The rules you are about to study are not arbitrary — they reflect the mathematical structure of the operations themselves, and they are essential knowledge for the IBEW Electrical Training Alliance Aptitude Test.

Core Principles & Definitions

The order of operations is a hierarchy of priority levels that tells you which calculations to carry out first when simplifying or evaluating a mathematical expression. The standard mnemonic used in the United States is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right). It is critical to understand that multiplication and division share the same priority level, as do addition and subtraction — you resolve ties by working from left to right across the expression.

Visual Explanation — The PEMDAS Hierarchy

Study the pyramid above carefully. The key insight is that parentheses are the ultimate override — anything enclosed in grouping symbols is evaluated first, regardless of what operations appear. Exponents come next because raising a number to a power is a more "compact" operation than multiplication, and multiplication in turn takes priority over addition because it represents repeated addition. This hierarchy is logical, not arbitrary, and understanding the reasoning behind it makes the rules far easier to remember and apply under test conditions.

Mathematical Framework

While the order of operations is a convention rather than a theorem, it can be stated precisely using a formal evaluation hierarchy. The following expressions illustrate how the rules dictate evaluation sequence and how violating them produces incorrect results.

On the IBEW aptitude test, expressions often include nested parentheses and fraction bars. Remember that a fraction bar acts as a grouping symbol — the entire numerator and the entire denominator are each evaluated before the division is performed. Similarly, when parentheses are nested, you begin with the innermost set and work outward.

Step-by-Step Evaluation Flowchart

When faced with a complex expression — especially under the time pressure of an aptitude test — it helps to follow a systematic procedure. The flowchart below provides a decision-making process that mirrors what experienced test-takers do instinctively. By training yourself to scan for each operation type in the correct order, you can avoid careless mistakes even when expressions look intimidating.

Worked Example

Let's walk through a multi-step expression that incorporates every level of the PEMDAS hierarchy. This type of problem is representative of what you might encounter on the IBEW aptitude test.

Evaluate: 5 + 3 × (8 − 2)² ÷ 9

Common Pitfalls & How to Avoid Them

Even students who know the rules of PEMDAS can stumble on certain expression structures that are designed to exploit common errors. The table below catalogs the most frequent mistakes and contrasts them with the correct approach. Recognizing these traps before test day will save you valuable time and prevent careless point losses.

Connecting to Advanced Algebra

The order of operations is not merely a test-prep topic — it is the backbone of every algebraic manipulation you will encounter in your electrical training. When you solve Ohm's Law for an unknown variable, simplify a load-calculation formula, or substitute values into the NEC voltage-drop equation, every step depends on correctly applying PEMDAS. The table below shows how basic order-of-operations skills connect to more advanced algebraic contexts you will meet as an apprentice.

As you advance through your electrical training, you will encounter increasingly complex formulas involving multiple grouping symbols, exponents, and nested fractions. The good news is that the PEMDAS framework never changes — it simply gets applied to more complicated expressions. Mastering the fundamentals now means you will have a reliable mental framework for every calculation you face in the field, from conductor sizing to transformer tap adjustments.

Practice Problems

Work through the following five problems in order, applying the PEMDAS framework you've learned. Each problem increases in complexity. After attempting each one, check your work against the provided solution.

Lesson Summary