2.1 Order of Operations, Signed Numbers & Exponents
Key Takeaways
- The standard order of operations (PEMDAS/GEMS) dictates evaluating Parentheses/Grouping symbols first, Exponents second, Multiplication and Division left-to-right third, and Addition and Subtraction left-to-right fourth.
- Operations with signed numbers require strict adherence to sign rules: multiplying or dividing numbers with identical signs yields positive results, while unlike signs yield negative results.
- Fundamental laws of exponents—including Product Rule ($a^m \cdot a^n = a^{m+n}$), Quotient Rule ($a^m / a^n = a^{m-n}$), Power of a Power ($(a^m)^n = a^{mn}$), and Zero Exponent ($a^0 = 1$)—enable rapid expression simplification.
- In electrical applications, power dissipation follows Joule's Law ($P = I^2 R$); because current $I$ is squared, doubling the circuit current quadruples thermal power loss.
- Calculators are prohibited on the EIAT; candidates must rely on mental arithmetic, factoring shortcuts, and estimation techniques to execute calculations.
2.1 Order of Operations, Signed Numbers & Exponents
Quick Summary: Mastering order of operations (PEMDAS), operations with signed positive and negative numbers, and laws of exponents is fundamental to scoring high on Part 1 of the EIAT. Because calculators are prohibited, candidates must execute mathematical operations in exact sequence, handle negative signs without error, and apply exponent laws ($a^m \cdot a^n = a^{m+n}$) instantly when evaluating electrical formulas such as Joule's Law of Power Dissipation ($P = I^2 R$).
Order of Operations: The PEMDAS Framework
In algebra and technical trades, mathematical expressions must be evaluated in a strict, standardized sequence. Without a universal hierarchy of operations, a single numerical expression could yield multiple conflicting answers. The standard order of operations is governed by the acronym PEMDAS (frequently remembered as Please Excuse My Dear Aunt Sally) or GEMS (Grouping, Exponents, Multiplication/Division, Subtraction/Addition).
The Standard Operational Hierarchy
- P - Parentheses & Grouping Symbols: Evaluate all operations enclosed within grouping symbols first. Grouping symbols include standard parentheses $( )$, square brackets $[ ]$, curly braces ${ }$, absolute value bars $| |$, and fractional division bars where the entire numerator or denominator acts as an implicit group. Work from the innermost set of parentheses outward.
- E - Exponents & Radicals: Calculate all powers (exponents) and roots (square roots, cube roots) from left to right.
- MD - Multiplication & Division (Equal Precedence): Perform all multiplication and division in strict order as they appear from left to right. Multiplication does not take precedence over division; if division appears to the left of multiplication, execute division first.
- AS - Addition & Subtraction (Equal Precedence): Perform all addition and subtraction in strict order as they appear from left to right. Like multiplication and division, addition does not take precedence over subtraction.
PEMDAS Operational Hierarchy
Level 1: ( ) [ ] { } ---> Parentheses & Grouping (Innermost First)
Level 2: x² √x ---> Exponents & Roots (Left to Right)
Level 3: × ÷ ---> Multiplication & Division (Left to Right)
Level 4: + - ---> Addition & Subtraction (Left to Right)
Common Traps in Order of Operations
On the EIAT, questions are intentionally engineered to test whether candidates succumb to common operational pitfalls:
- Left-to-Right Violation in Multiplication/Division: In the expression $12 \div 3 \times 2$, performing multiplication first yields $12 \div 6 = 2$, which is incorrect. Because division and multiplication share equal precedence, evaluating left to right yields $4 \times 2 = 8$.
- Negation vs. Exponentiation: The expression $-3^2$ means $-(3 \times 3) = -9$, because exponentiation precedes negation. However, $(-3)^2$ means $(-3) \times (-3) = +9$, because the parenthesis forces the negative sign to be squared.
Arithmetic Rules for Signed Numbers
Working with positive and negative numbers is an indispensable skill in electrical theory, where direction of current, voltage potential differences, and power vector phase angles rely on signed quantities.
Addition and Subtraction of Signed Numbers
- Adding Same Signs: Add the absolute values of the numbers and keep the common sign.
- Adding Different Signs: Subtract the smaller absolute value from the larger absolute value, and attach the sign of the number with the larger absolute value.
- Subtracting Signed Numbers: Change the subtraction operation to addition and invert the sign of the number being subtracted (add the opposite).
Multiplication and Division of Signed Numbers
- Like Signs Yield Positive Results: Multiplying or dividing two numbers with the same sign (both positive or both negative) always results in a positive value.
- Unlike Signs Yield Negative Results: Multiplying or dividing two numbers with different signs (one positive and one negative) always results in a negative value.
Fundamental Laws of Exponents
An exponent indicates how many times a base number is multiplied by itself. For an expression $a^n$, $a$ is the base and $n$ is the exponent (or power). Mastering exponent rules allows rapid simplification of algebraic expressions without performing longhand calculations.
Summary of Key Exponent Rules
| Exponent Rule | General Formula | Concrete Numerical Example |
|---|---|---|
| Product Law | $a^m \cdot a^n = a^{m+n}$ | $x^3 \cdot x^4 = x^{3+4} = x^7$ |
| Quotient Law | $\frac{a^m}{a^n} = a^{m-n}$ ($a \neq 0$) | $\frac{y^8}{y^3} = y^{8-3} = y^5$ |
| Power of a Power | $(a^m)^n = a^{m \cdot n}$ | $(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64$ |
| Power of a Product | $(ab)^n = a^n b^n$ | $(3x)^3 = 3^3 \cdot x^3 = 27x^3$ |
| Power of a Quotient | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ ($b \neq 0$) | $\left(\frac{x}{4}\right)^2 = \frac{x^2}{4^2} = \frac{x^2}{16}$ |
| Zero Exponent Rule | $a^0 = 1$ ($a \neq 0$) | $125^0 = 1 \quad \text{and} \quad (-9)^0 = 1$ |
| Negative Exponent Rule | $a^{-n} = \frac{1}{a^n}$ ($a \neq 0$) | $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ |
Electrical Trade Application: Power Dissipation & Joule's Law
In electrical work, exponent laws and signed numbers are applied directly in power calculations. Joule's Law of Heating states that the power $P$ dissipated as heat in an electrical conductor is proportional to the square of the current $I$ flowing through it multiplied by the resistance $R$:
Where:
- $P$ = Power dissipation in Watts ($\text{W}$)
- $I$ = Electric current in Amperes ($\text{A}$)
- $R$ = Resistance in Ohms ($\Omega$)
The Non-Linear Impact of Current Squaring
Because current $I$ is squared ($I^2$), doubling the current flowing through a conductor does not simply double the heat loss—it quadruples the power dissipation!
- If current $I = 5\text{ A}$ and resistance $R = 10\ \Omega$:
- If current doubles to $I = 10\text{ A}$ through the same $10\ \Omega$ resistance:
Understanding this exponential relationship helps electricians comprehend why commercial power distribution systems step up transmission line voltages to high levels—reducing current $I$ minimizes severe $I^2 R$ power losses in line conductors over long distances.
Step-by-Step Worked Examples
Worked Example 1: Complex Order of Operations with Exponents
Problem: Evaluate the expression:
Solution:
- Parentheses First: Evaluate the term inside the parenthesis $(5 - 2)$:
- Exponents Second: Evaluate both exponential terms $(3)^2$ and $(-2)^3$:
- Multiplication & Division Left-to-Right:
- Execute multiplication: $4 \times 9 = 36$
- Execute division: $16 \div (-8) = -2$
- Addition & Subtraction Left-to-Right:
- Subtract: $24 - 36 = -12$
- Add signed number: $-12 + (-2) = -14$
Final Answer: $-14$
Worked Example 2: Simplifying Expressions with Laws of Exponents
Problem: Simplify the algebraic expression completely:
Solution:
- Multiply Numerator Coefficients and Combine Variable Exponents using Product Law ($a^m \cdot a^n = a^{m+n}$):
- Divide Coefficient and Apply Quotient Law ($\frac{a^m}{a^n} = a^{m-n}$) to Variables:
- Simplify Coefficients and Subtract Powers:
Final Answer: $4x^5 y^4$
Worked Example 3: Electrical Power Dissipation Calculation
Problem: A 240-Volt industrial heating element has a resistance of $8\ \Omega$. If the current flowing through the element is $30\text{ A}$, calculate the total thermal power dissipation in Watts ($\text{W}$) and Kilowatts ($\text{kW}$).
Solution:
- Identify Given Values and Select Power Formula:
- Calculate Exponent ($I^2$):
- Multiply by Resistance ($R$):
- Convert Watts to Kilowatts ($1\text{ kW} = 1,000\text{ W}$):
Final Answer: $7,200\text{ W}$ (or $7.2\text{ kW}$)
Evaluate the expression: $18 - 3 \times (4 - 2)^2 + 6 \div 2$
An electrician is calculating power dissipation using $P = I^2 R$. If current $I = 10\text{ A}$ and resistance $R = 6\ \Omega$, what is the power dissipation?
Simplify the expression completely using laws of exponents: $(4a^3 b) \times (6a^2 b^3) \div (12b^2)$