5.1 Number Properties, Exponents, Radicals, and Polynomials

5.1 Number Properties, Exponents, Radicals, and Polynomials

Core Principle: The Mathematics Knowledge (MK) subtest assesses high school-level mathematical principles, theoretical rules, and computational procedures. On the computerized CAT-ASVAB, you have 15 scored adaptive questions in 31 minutes — giving you roughly 124 seconds per question, about twice the allowance of the technical recall subtests. Unlike Arithmetic Reasoning (AR), which evaluates word problems and practical applied scenarios, MK focuses on pure mathematical mechanics: evaluating complex expressions, simplifying radicals, manipulating exponents, factoring polynomials, and solving algebraic systems without a calculator.

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1. CAT-ASVAB Mathematics Knowledge (MK) Architecture & Tactical Pacing

Mathematics Knowledge is one of the four academic subtests that determine the Armed Forces Qualification Test (AFQT) percentile score, which dictates overall military enlistment eligibility:

$\text{AFQT Raw Score} = 2\text{VE} + \text{AR} + \text{MK}$

Where $\text{VE} = \text{WK} + \text{PC}$ (Verbal Expression). While VE carries double weight, MK and AR together comprise 50% of the non-verbal AFQT score. Furthermore, MK is a mandatory composite component for high-tier technical military occupational specialties (MOS/Ratings/AFSCs):

• Skilled Technical (ST): $\text{GS} + \text{VE} + \text{MK} + \text{MC}$ (Required for Army Cyber Operations, Intelligence, and IT)
• Electronics (EL): $\text{GS} + \text{AR} + \text{MK} + \text{EI}$ (Required for Navy Nuclear Field, Air Force Avionics, and Submarine Warfare)
• General Maintenance (GM): $\text{GS} + \text{AS} + \text{MK} + \text{EI}$ (Prerequisite for Aviation Maintenance and Armament)
• General Technical (GT): $\text{VE} + \text{AR}$ (Officer Candidate School and Special Operations prerequisite)

Non-Calculator Execution & Time Management

Calculators are strictly prohibited during the ASVAB. Candidates are issued scratch paper and a pencil at the Military Entrance Processing Station (MEPS). Excelling on the MK subtest demands fluent mental arithmetic, disciplined scratchpad layout, and instant recognition of structural algebraic shortcuts.

+-----------------------------------------------------------------------------------------+
|                               MK TIMING BUDGET: 75 SECONDS                              |
+-----------------------+-------------------------+---------------------------------------+
| 0 - 15 Seconds        | 15 - 55 Seconds         | 55 - 75 Seconds                       |
| Classify concept and  | Execute algebraic steps | Verify signs, check exponents, and    |
| identify shortcut     | on scratch paper        | confirm selected multiple-choice item |
+-----------------------+-------------------------+---------------------------------------+

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2. Classification of Real Numbers & Foundational Arithmetic Properties

All numbers encountered on the MK subtest belong to the set of Real Numbers ($\mathbb{R}$):

[ Real Numbers (R) ]
  ├── [ Rational Numbers (Q) ] (can be expressed as a/b, where a, b ∈ Z, b ≠ 0)
  │     ├── [ Integers (Z) ] (..., -3, -2, -1, 0, 1, 2, 3, ...)
  │     │     └── [ Whole Numbers (W) ] (0, 1, 2, 3, ...)
  │     │           └── [ Natural Numbers (N) ] (1, 2, 3, ...)
  │     └── [ Fractions & Decimals ] (Terminating e.g. 0.375, Repeating e.g. 0.666...)
  └── [ Irrational Numbers ] (Cannot be written as a/b; non-terminating, non-repeating: √2, √3, π, e)

Critical Number Definitions:

• Prime Numbers: Integers strictly greater than 1 whose only positive integer divisors are 1 and themselves. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Note that 2 is the only even prime number, and 1 is neither prime nor composite.
• Composite Numbers: Positive integers greater than 1 that possess at least one positive factor other than 1 and themselves (e.g., 4, 6, 8, 9, 10, 12, 14, 15...).
• Reciprocals (Multiplicative Inverses): Two non-zero numbers whose product is exactly 1. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. The integer $n$ has reciprocal $\frac{1}{n}$. Zero has no reciprocal.
• Absolute Value ($|x|$): The non-negative geometric distance of a number from zero on the real number line ($|-9| = 9$, $|9| = 9$).

Fundamental Arithmetic Properties

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3. Rapid Divisibility Tests, Prime Factorization, GCF & LCM

Divisibility Diagnostic Table

Mastering divisibility tests allows instantaneous reduction of large fractional coefficients on scratch paper:

Prime Factorization & Finding GCF and LCM

According to the Fundamental Theorem of Arithmetic, every composite number factors uniquely into a product of prime powers.

Prime Factorizations for 48, 72, and 120:
       48 = 16 × 3 = 2⁴ × 3¹
       72 = 8 × 9  = 2³ × 3²
      120 = 8 × 15 = 2³ × 3¹ × 5¹

1. Greatest Common Factor (GCF): Take the lowest exponent of every prime factor common to all sets: $\text{GCF}(48, 72, 120) = 2^{\min(4,3,3)} \times 3^{\min(1,2,1)} = 2^3 \times 3^1 = 8 \times 3 = 24$
2. Least Common Multiple (LCM): Take the highest exponent of every prime factor present in any set: $\text{LCM}(48, 72, 120) = 2^{\max(4,3,3)} \times 3^{\max(1,2,1)} \times 5^{\max(0,0,1)} = 2^4 \times 3^2 \times 5^1 = 16 \times 9 \times 5 = 720$

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4. Order of Operations (PEMDAS / GEMS) & Signed Numbers

When evaluating algebraic expressions, operations must strictly follow hierarchical priority:

[ P / G ] : Grouping Symbols (Parentheses (), Brackets [], Braces {}, Numerator/Denominator bars)
[   E   ] : Exponents and Radicals (evaluated left-to-right)
[ M / D ] : Multiplication & Division (Equal priority: evaluate Left-to-Right)
[ A / S ] : Addition & Subtraction (Equal priority: evaluate Left-to-Right)

ASVAB Trap: Multiplication does not outrank division! They have identical priority and must be executed in left-to-right sequence. For example: $18 \div 6 \times 3 = 3 \times 3 = 9$, NOT $18 \div 18 = 1$.

Exponent Sign Traps with Negative Bases

• Base enclosed in parentheses: $(-4)^2 = (-4) \times (-4) = +16$
• Base without parentheses: $-4^2 = -(4 \times 4) = -16$
• Odd power with negative base: $(-2)^3 = (-2) \times (-2) \times (-2) = -8$

Step-by-Step Worked Solution:

$\text{Evaluate: } 24 - 3 \cdot [4 + (2 - 6)^2] \div (-5 + 1)$

1. Resolve inner grouping: $2 - 6 = -4$ and $-5 + 1 = -4$
2. Expression simplifies to: $24 - 3 \cdot [4 + (-4)^2] \div (-4)$
3. Resolve exponent inside bracket: $(-4)^2 = 16$
4. Add inside bracket: $4 + 16 = 20$
5. Expression simplifies to: $24 - 3 \cdot 20 \div (-4)$
6. Execute multiplication/division left-to-right: $3 \cdot 20 = 60$, then $60 \div (-4) = -15$
7. Final subtraction: $24 - (-15) = 24 + 15 = 39$

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5. Rules of Exponents, Radicals & Scientific Notation

Laws of Exponents Reference Table

Simplifying Radicals

To simplify square roots without a calculator, factor out the largest perfect square ($1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...$): $\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}$

Combining and Rationalizing Radicals

• Combining Like Radicals: $4\sqrt{18} - 2\sqrt{50} + \sqrt{72} = 4(3\sqrt{2}) - 2(5\sqrt{2}) + 6\sqrt{2} = 12\sqrt{2} - 10\sqrt{2} + 6\sqrt{2} = 8\sqrt{2}$
• Rationalizing Denominators: Multiply numerator and denominator by the radical to eliminate square roots from the denominator: $\frac{15}{\sqrt{5}} = \frac{15 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{15\sqrt{5}}{5} = 3\sqrt{5}$

Scientific Notation Operations

• Multiplication: $(2.5 \times 10^4) \times (6.0 \times 10^3) = (2.5 \times 6.0) \times 10^{4+3} = 15.0 \times 10^7 = 1.5 \times 10^8$
• Division: $\frac{7.2 \times 10^6}{1.8 \times 10^{-2}} = \left(\frac{7.2}{1.8}\right) \times 10^{6 - (-2)} = 4.0 \times 10^8$

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6. Polynomial Operations, Special Products & Factoring Archetypes

Definitions:

• Monomial: Single term (e.g., $7x^3$).
• Binomial: Two terms separated by $+$ or $-$ (e.g., $4x^2 - 9$).
• Trinomial: Three terms (e.g., $2x^2 + 5x - 12$).
• Degree: The highest exponent of the variable in the polynomial.

Polynomial Multiplication: The FOIL Method

To multiply two binomials $(ax + b)(cx + d)$, apply FOIL (First, Outer, Inner, Last):

          (3x - 4)(2x + 5)
          ├── First:  3x · 2x  = 6x²
          ├── Outer:  3x · 5   = +15x
          ├── Inner:  (-4) · 2x = -8x
          └── Last:   (-4) · 5  = -20
          Sum = 6x² + (15x - 8x) - 20 = 6x² + 7x - 20

Special Algebraic Products

• Difference of Squares: $(a - b)(a + b) = a^2 - b^2$
• Square of a Binomial: $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$

5 Core ASVAB Factoring Archetypes

+-----------------------------------------------------------------------------------------+
|                             5 ASVAB FACTORING ARCHETYPES                                |
+------------------------------------+----------------------------------------------------+
| 1. Greatest Common Factor (GCF)    | 12x⁴y² - 18x²y³ = 6x²y²(2x² - 3y)                 |
+------------------------------------+----------------------------------------------------+
| 2. Difference of Two Squares       | a² - b² = (a - b)(a + b)                           |
|                                    | 49x² - 36 = (7x - 6)(7x + 6)                       |
+------------------------------------+----------------------------------------------------+
| 3. Perfect Square Trinomials       | a² ± 2ab + b² = (a ± b)²                           |
|                                    | 9x² - 24x + 16 = (3x - 4)²                         |
+------------------------------------+----------------------------------------------------+
| 4. General Trinomials (ax²+bx+c)   | 2x² + 7x + 3 = (2x + 1)(x + 3)                      |
+------------------------------------+----------------------------------------------------+
| 5. Factoring by Grouping           | x³ + 2x² + 5x + 10 = x²(x + 2) + 5(x + 2)          |
|                                    |                     = (x² + 5)(x + 2)             |
+------------------------------------+----------------------------------------------------+

ASVAB Trap: The sum of two squares ($a^2 + b^2$) is prime and cannot be factored over the set of real numbers! For example, $x^2 + 25$ cannot be factored into $(x+5)(x+5)$ or $(x-5)(x+5)$.