6.1 Rational Numbers, Absolute Value, and Order of Operations

Rational Numbers, Order of Operations, Exponents & Radicals

Core Foundation: The Next-Generation Quantitative Reasoning, Algebra, and Statistics (QAS) assessment measures your proficiency in manipulating real numbers, applying arithmetic hierarchies, executing exponent operations, and simplifying radical structures. Precision with signed values, negative exponents, and grouping boundaries is essential for every algebraic domain tested on the ACCUPLACER.

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The Real Number System Hierarchy

All numbers evaluated on the ACCUPLACER QAS assessment reside within the set of Real Numbers ($\mathbb{R}$). Understanding the classification of numbers provides the mathematical foundation for evaluating algebraic constraints, domains, and arithmetic properties.

+-------------------------------------------------------------------------+
|                         THE REAL NUMBER SYSTEM (ℝ)                      |
+-------------------------------------------------------------------------+
|  RATIONAL NUMBERS (ℚ)                     │  IRRATIONAL NUMBERS         |
|  Any number expressible as a/b,          │  Non-terminating,           |
|  where a, b ∈ ℤ and b ≠ 0.                │  non-repeating decimals.    |
|  (e.g., -7, 0, 3/4, 0.625, 0.333...)      │  (e.g., √2, √5, π, e)       |
|                                           │                             |
|   ┌─────────────────────────────────────┐ │                             |
|   │ INTEGERS (ℤ)                        │ │                             |
|   │ {..., -3, -2, -1, 0, 1, 2, 3, ...}  │ │                             |
|   │   ┌───────────────────────────────┐ │ │                             |
|   │   │ WHOLE NUMBERS (𝕎)             │ │ │                             |
|   │   │ {0, 1, 2, 3, ...}             │ │ │                             |
|   │   │   ┌─────────────────────────┐ │ │ │                             |
|   │   │   │ NATURAL NUMBERS (ℕ)     │ │ │ │                             |
|   │   │   │ {1, 2, 3, 4, ...}       │ │ │ │                             |
|   │   │   └─────────────────────────┘ │ │ │                             |
|   │   └───────────────────────────────┘ │ │                             |
|   └─────────────────────────────────────┘ │                             |
+-------------------------------------------------------------------------+

Sets of Real Numbers Defined

1. Natural Numbers ($\mathbb{N}$): Counting numbers starting from positive one: $\{1, 2, 3, 4, \dots\}$.
2. Whole Numbers ($\mathbb{W}$): Natural numbers combined with zero: $\{0, 1, 2, 3, \dots\}$.
3. Integers ($\mathbb{Z}$): Positive and negative whole numbers along with zero: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
4. Rational Numbers ($\mathbb{Q}$): Numbers expressible in the fractional form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. In decimal form, rational numbers either terminate (e.g., $\frac{3}{8} = 0.375$) or repeat infinitely in a predictable cycle (e.g., $\frac{4}{11} = 0.\overline{36}$).
5. Irrational Numbers: Real numbers that cannot be represented as an integer ratio. Their decimal expansions are non-terminating and non-repeating (e.g., $\sqrt{2} \approx 1.4142\dots$, $\sqrt{7} \approx 2.6457\dots$, $\pi \approx 3.14159\dots$).

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Operations with Signed Numbers & Absolute Value

Arithmetic errors on the ACCUPLACER most commonly stem from sign mismanagement during multi-step algebraic evaluations.

Rules for Signed Arithmetic

• Addition:
  • Same Signs: Add the absolute values and retain the shared sign (e.g., $-8 + (-5) = -13$).
  • Different Signs: Subtract the smaller absolute value from the larger absolute value and retain the sign of the number with the larger absolute value (e.g., $-14 + 9 = -5$; $18 + (-7) = 11$).
• Subtraction (Adding the Additive Inverse):
  • Convert subtraction into addition of the opposite: $a - b = a + (-b)$.
  • Handling double negatives: $a - (-b) = a + b$ (e.g., $-7 - (-12) = -7 + 12 = 5$).
• Multiplication and Division:
  • Same Signs (Positive $\times$ Positive or Negative $\times$ Negative): Result is always positive ($(-6) \times (-4) = +24$; $\frac{-30}{-5} = +6$).
  • Different Signs (Positive $\times$ Negative or Negative $\times$ Positive): Result is always negative ($(-7) \times 3 = -21$; $\frac{48}{-6} = -8$).

Absolute Value and Absolute Value Equations

The absolute value of a real number $x$, denoted $|x|$, represents its undirected geometric distance from zero on the real number line. Because distance cannot be negative, $|x| \ge 0$ for all $x \in \mathbb{R}$.

$$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

When solving absolute value equations of the form $|ax + b| = c$:

1. If $c < 0$, there is no real solution ($|ax + b| = -4$ is impossible).
2. If $c = 0$, solve the single equation $ax + b = 0$.
3. If $c > 0$, split into two distinct linear cases: $$\text{Case 1: } ax + b = c \quad \text{or} \quad \text{Case 2: } ax + b = -c$$

Worked Example: Solving an Absolute Value Equation

Solve for $x$: $4|2x - 3| - 7 = 13$.

• Step 1 (Isolate the absolute value term): Add 7 to both sides: $4|2x - 3| = 20$. Divide both sides by 4: $|2x - 3| = 5$.
• Step 2 (Split into two cases):
  • Case 1: $2x - 3 = 5 \implies 2x = 8 \implies x = 4$
  • Case 2: $2x - 3 = -5 \implies 2x = -2 \implies x = -1$
• Step 3 (Verify solutions):
  • For $x = 4$: $4|2(4) - 3| - 7 = 4|5| - 7 = 20 - 7 = 13$ (Valid).
  • For $x = -1$: $4|2(-1) - 3| - 7 = 4|-5| - 7 = 20 - 7 = 13$ (Valid).
  • The solution set is $\{ -1, 4 \}$.

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The Order of Operations (PEMDAS / GEMS)

To evaluate complex arithmetic and algebraic expressions unambiguously, mathematics adheres to a standardized hierarchy of operations:

  P / G : Parentheses & Grouping Symbols [ ( ), [ ], { }, | |, √( ), fraction bars ]
  E     : Exponents & Radicals (Powers, Square Roots, Indices)
  M / D : Multiplication & Division (Evaluated STRICTLY Left-to-Right)
  A / S : Addition & Subtraction (Evaluated STRICTLY Left-to-Right)

Critical Operational Pitfalls on the ACCUPLACER

1. The Left-to-Right Rule for Division and Multiplication: Multiplication does not outrank division. They share equal priority and must be evaluated in order of appearance from left to right. $$\text{Example: } 24 \div 4 \times 2 = 6 \times 2 = 12 \quad (\text{NOT } 24 \div 8 = 3)$$
2. The Left-to-Right Rule for Subtraction and Addition: Addition does not outrank subtraction. $$\text{Example: } 15 - 8 + 4 = 7 + 4 = 11 \quad (\text{NOT } 15 - 12 = 3)$$
3. The Unary Negation vs. Base Exponent Trap:
   • $(-3)^2 = (-3) \times (-3) = +9$ (The base is $-3$).
   • $-3^2 = -(3^2) = -(9) = -9$ (The base is $3$; exponentiation occurs before applying the negative sign).
4. Fraction Bars as Implicit Grouping Symbols: In expressions such as $\frac{18 - 6}{2 + 4}$, the numerator and denominator constitute separate, enclosed grouping envelopes. Evaluate the entire numerator ($18 - 6 = 12$) and entire denominator ($2 + 4 = 6$) before executing the division: $\frac{12}{6} = 2$.

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The Laws of Integer & Rational Exponents

Exponents denote repeated multiplication: $x^n = \underbrace{x \cdot x \cdot x \dots x}_{n \text{ times}}$. Algebraic expressions on the QAS assessment require fluent application of the fundamental exponent laws.

Exponent Laws Reference Table

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Scientific Notation Operations

Scientific notation formats very large or very small real numbers into standard form:

$$a \times 10^b \quad \text{where } 1 \le |a| < 10 \text{ and } b \in \mathbb{Z}$$

Arithmetic Operations in Scientific Notation

• Multiplication: Multiply the decimal coefficients and add the powers of 10: $$(3.0 \times 10^5) \times (4.0 \times 10^3) = (3.0 \times 4.0) \times 10^{5+3} = 12.0 \times 10^8 = 1.2 \times 10^9$$
• Division: Divide the decimal coefficients and subtract the powers of 10: $$\frac{1.6 \times 10^3}{4.0 \times 10^7} = \left(\frac{1.6}{4.0}\right) \times 10^{3-7} = 0.4 \times 10^{-4} = 4.0 \times 10^{-5}$$
• Addition / Subtraction: Adjust terms so that both exponents match before adding or subtracting coefficients: $$(5.2 \times 10^4) + (3.1 \times 10^3) = (5.2 \times 10^4) + (0.31 \times 10^4) = (5.2 + 0.31) \times 10^4 = 5.51 \times 10^4$$

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Simplifying Square Roots & Radicals

The principal square root of a non-negative real number $x$, denoted $\sqrt{x}$, is the unique non-negative number $y$ such that $y^2 = x$.

Fundamental Radical Properties

1. Product Property of Radicals: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ (for $a, b \ge 0$)
2. Quotient Property of Radicals: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (for $a \ge 0, b > 0$)

Simplifying Radicals by Factoring Perfect Squares

To simplify $\sqrt{N}$, identify the largest perfect square factor ($1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \dots$) dividing $N$:

$$\sqrt{180} = \sqrt{36 \cdot 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5}$$ $$\sqrt{72x^5} = \sqrt{36 \cdot 2 \cdot x^4 \cdot x} = 6x^2\sqrt{2x}$$

Radical Arithmetic (Addition, Subtraction & Multiplication)

• Adding/Subtracting Like Radicals: Only radicals with identical indices and identical radicands can be combined: $$4\sqrt{18} - \sqrt{50} + 3\sqrt{8} = 4(3\sqrt{2}) - (5\sqrt{2}) + 3(2\sqrt{2}) = 12\sqrt{2} - 5\sqrt{2} + 6\sqrt{2} = 13\sqrt{2}$$
• Multiplying Radicals: Multiply coefficients together and radicands together: $$(2\sqrt{6})(5\sqrt{3}) = 10\sqrt{18} = 10(3\sqrt{2}) = 30\sqrt{2}$$
• Distributing Binomials with Radicals (FOIL): $$(3 + \sqrt{5})(2 - \sqrt{5}) = 6 - 3\sqrt{5} + 2\sqrt{5} - (\sqrt{5})^2 = 6 - \sqrt{5} - 5 = 1 - \sqrt{5}$$

Rationalizing Denominators

A radical expression is not in simplest form if a radical remains in the denominator.

• Single Radical Denominator: Multiply numerator and denominator by the radical: $$\frac{12}{\sqrt{6}} = \frac{12}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{12\sqrt{6}}{6} = 2\sqrt{6}$$
• Binomial Radical Denominator (Conjugate Multiplication): The conjugate of $(a + \sqrt{b})$ is $(a - \sqrt{b})$, using $(x+y)(x-y) = x^2 - y^2$: $$\frac{6}{3 - \sqrt{3}} = \frac{6(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} = \frac{18 + 6\sqrt{3}}{3^2 - (\sqrt{3})^2} = \frac{18 + 6\sqrt{3}}{9 - 3} = \frac{18 + 6\sqrt{3}}{6} = 3 + \sqrt{3}$$