Overview
The ASVAB Mathematics Knowledge section tests your ability to apply core math concepts including algebra, geometry, number theory, and exponents. This guide covers essential formulas, rules, and problem-solving strategies drawn directly from high-frequency flashcard topics. Mastering these concepts will help you maximize your ASVAB score and qualify for a wider range of military career opportunities.
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Algebra
Core Concepts
Algebra involves solving equations and inequalities, working with functions, and manipulating expressions. The primary goal is isolating a variable to find its value.
Solving Linear Equations
• Isolate the variable by performing inverse operations on both sides
• Work in reverse PEMDAS order: undo addition/subtraction first, then multiplication/division
• Example: 3x + 9 = 24 → subtract 9 → 3x = 15 → divide by 3 → x = 5
• Example: 4(x − 2) = 2(x + 6) → distribute → 4x − 8 = 2x + 12 → 2x = 20 → x = 10
Key Forms & Formulas
| Concept | Formula/Rule |
|---|---|
| Slope-Intercept Form | y = mx + b |
| Quadratic Formula | x = (−b ± √(b² − 4ac)) / 2a |
| FOIL Method | (a + b)(c + d) = ac + ad + bc + bd |
| Difference of Squares | (a + b)(a − b) = a² − b² |
Functions
• To evaluate f(x), substitute the given value for every x in the expression
• Example: f(x) = 2x² − 3x + 1, find f(2): 2(4) − 3(2) + 1 = 8 − 6 + 1 = 3
Factoring Polynomials
• Factoring rewrites a polynomial as a product of simpler expressions
• Always look for the Greatest Common Factor (GCF) first
• Recognize special patterns like the difference of squares: x² − 9 = (x + 3)(x − 3)
Systems of Equations (Substitution Method)
1. Solve one equation for one variable
2. Substitute that expression into the second equation
3. Solve for the remaining variable
4. Back-substitute to find the first variable
Inequalities
• Use the same rules as equations, except:
• ⚠️ Flip the inequality sign when multiplying or dividing both sides by a negative number
• Example: −2x > 6 → x < −3 (sign flips!)
Key Terms — Algebra
• Variable – A symbol (usually x or y) representing an unknown value
• Coefficient – The number multiplied by a variable (e.g., 3 in 3x)
• Binomial – A polynomial with exactly two terms
• Root/Solution – The value(s) of x that satisfy an equation
• Slope (m) – The rate of change of a line (rise over run)
• Y-intercept (b) – Where a line crosses the y-axis
⚠️ Watch Out For — Algebra
• Distributing negatives: In 4(x − 2), don't forget to distribute the 4 to BOTH terms: 4x − 8, not 4x − 2
• FOIL vs. difference of squares: Recognize (x + 3)(x − 3) as a shortcut pattern rather than using full FOIL
• Quadratic formula sign errors: The formula uses −b, so if b is already negative, −b becomes positive
• Inequality direction: The flip rule applies ONLY to multiplication/division by negatives, not addition/subtraction
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Geometry
Core Concepts
Geometry covers shapes, angles, measurements, and spatial relationships. For the ASVAB, focus on memorizing key formulas and understanding when to apply them.
Angle Relationships
| Type | Definition |
|---|---|
| Complementary Angles | Sum = 90° |
| Supplementary Angles | Sum = 180° |
| Triangle Interior Angles | Sum = 180° |
| Quadrilateral Interior Angles | Sum = 360° |
Triangle Formulas
• Area of a Triangle: A = ½ × base × height
- The height must be perpendicular to the base
• Pythagorean Theorem: a² + b² = c²
- Only applies to right triangles
- c is always the hypotenuse (longest side, opposite the right angle)
- Example: legs 6 and 8 → 36 + 64 = 100 → c = 10 (memorize: 3-4-5 and 6-8-10 triangles)
Circle Formulas
• Circumference: C = 2πr = πd
• Area: A = πr²
• Remember: diameter = 2 × radius
3D Shape Formulas
• Volume of a Rectangular Prism: V = l × w × h
• Surface Area of a Cube: SA = 6s²
- A cube has 6 identical square faces, each with area s²
Key Terms — Geometry
• Hypotenuse – The side opposite the right angle in a right triangle; always the longest side
• Radius – Distance from the center of a circle to its edge
• Diameter – Distance across a circle through its center (= 2r)
• Perpendicular – Meeting at a 90° angle
• Quadrilateral – Any four-sided polygon
⚠️ Watch Out For — Geometry
• Height vs. slant height: The height in A = ½bh must be the perpendicular height, not a slanted side
• Radius vs. diameter: Area uses r², circumference can use r or d — don't mix them up
• Pythagorean Theorem only works for right triangles — don't apply it to other triangles
• Common Pythagorean triples to memorize: 3-4-5, 5-12-13, 6-8-10, 8-15-17
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Number Theory & Properties
Core Concepts
Number theory covers the fundamental properties of numbers, including types of numbers, factors, multiples, and operational rules.
Types of Numbers
| Type | Definition | Examples |
|---|---|---|
| Prime | Exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13 |
| Composite | More than two factors | 4, 6, 8, 9, 12 |
| Rational | Expressible as p/q (integers, q ≠ 0) | ½, 0.75, −3, 7 |
| Irrational | Cannot be expressed as p/q | π, √2, √3 |
> Note: 1 is neither prime nor composite. 2 is the only even prime number.
GCF and LCM
• Greatest Common Factor (GCF): The largest factor shared by two numbers
- GCF(24, 36) = 12
- Use to simplify fractions and factor expressions
• Least Common Multiple (LCM): The smallest number both values divide into evenly
- LCM(4, 6) = 12
- Use to add/subtract fractions with different denominators
Order of Operations — PEMDAS
| Step | Operation |
|---|---|
| P | Parentheses |
| E | Exponents |
| MD | Multiplication & Division (left to right) |
| AS | Addition & Subtraction (left to right) |
Absolute Value & Signed Numbers
• Absolute value = distance from zero; always non-negative
- |−15| = 15, |15| = 15
• Sign rules for multiplication/division:
- Negative × Negative = Positive → (−3)(−4) = 12
- Negative × Positive = Negative → (−3)(4) = −12
- Positive × Positive = Positive
Key Terms — Number Theory
• Factor – A number that divides evenly into another number
• Multiple – The result of multiplying a number by any integer
• Integer – Any whole number (positive, negative, or zero)
• Absolute Value – The non-negative distance of a number from zero
⚠️ Watch Out For — Number Theory
• 1 is NOT prime — this is one of the most common errors on standardized tests
• PEMDAS left-to-right rule: Multiplication and division have equal priority and are solved left to right, not multiplication before division every time
• Negative × Negative = Positive: Don't forget this rule when simplifying expressions
• GCF vs. LCM confusion: GCF is for simplifying; LCM is for finding common denominators
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Exponents & Roots
Core Concepts
Exponents represent repeated multiplication; roots are the inverse operation. Both follow specific rules that must be memorized for efficient problem-solving.
Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x³ = x⁵ |
| Power Rule | (aᵐ)ⁿ = aᵐⁿ | (x³)² = x⁶ |
| Zero Exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional Exponent | a^(1/2) = √a | 4^(1/2) = 2 |
Fractional Exponents & Roots
• a^(1/2) = square root of a
• a^(1/3) = cube root of a
• a^(m/n) = (ⁿ√a)ᵐ
Common Values to Memorize
Perfect Squares:
• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Powers of 2:
• 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64
Square Roots:
• √144 = 12, √121 = 11, √169 = 13
Key Terms — Exponents & Roots
• Base – The number being multiplied (in 2⁵, the base is 2)
• Exponent/Power – Tells how many times to multiply the base by itself
• Radical – The √ symbol indicating a root
• Square Root – A number that, when multiplied by itself, gives the original number
• Reciprocal – The multiplicative inverse (1/x for a number x)
⚠️ Watch Out For — Exponents & Roots
• Zero exponent: Any nonzero base to the power of 0 equals 1, not 0
• Negative exponents: a⁻ⁿ means a reciprocal, NOT a negative number (2⁻² = 1/4, not −4)
• Power rule vs. product rule: (x³)² = x⁶ (multiply exponents); x³ · x² = x⁵ (add exponents)
• Fractional exponents: 4^(1/2) = √4 = 2, not 4/2 = 2 (coincidence — don't confuse the methods)
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Quick Review Checklist
Use this checklist to confirm you've mastered the essential concepts before test day:
Algebra
• [ ] Solve one- and two-step linear equations by isolating the variable
• [ ] Distribute and combine like terms correctly (including negatives)
• [ ] Apply the FOIL method and recognize the difference of squares pattern
• [ ] Use the quadratic formula to find roots of ax² + bx + c = 0
• [ ] Evaluate functions by substituting a given value
• [ ] Flip the inequality sign when multiplying/dividing by a negative
• [ ] Solve systems of equations using substitution
• [ ] Identify slope (m) and y-intercept (b) in y = mx + b
Geometry
• [ ] Calculate area of triangles (½bh) and circles (πr²)
• [ ] Calculate circumference (2πr or πd)
• [ ] Apply the Pythagorean Theorem (a² + b² = c²) to right triangles
• [ ] Recall angle sums: triangles = 180°, quadrilaterals = 360°
• [ ] Distinguish complementary (90°) from supplementary (180°) angles
• [ ] Calculate volume of rectangular prisms (lwh)
• [ ] Calculate surface area of a cube (6s²)
• [ ] Recognize common Pythagorean triples: 3-4-5, 6-8-10
Number Theory & Properties
• [ ] Identify prime numbers (exactly 2 factors; 1 is NOT prime)
• [ ] Find GCF and LCM of two numbers
• [ ] Apply PEMDAS correctly, including left-to-right for MD and AS
• [ ] Determine absolute value (always non-negative)
• [ ] Apply sign rules for multiplying/dividing negative numbers
• [ ] Distinguish rational from irrational numbers
Exponents & Roots
• [ ] Apply the product rule (add exponents) and power rule (multiply exponents)
• [ ] Simplify negative exponents as reciprocals
• [ ] Recall that any nonzero number to the zero power = 1
• [ ] Convert fractional exponents to radicals (a^(1/2) = √a)
• [ ] Memorize perfect squares up to 15² = 225 and powers of 2 up to 2⁶ = 64
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> 💡 Final Test-Day Tips:
> - Write out every step — careless errors on multi-step problems are the #1 source of lost points
> - Plug answers back in to verify solutions when time allows
> - Eliminate obviously wrong answers on multiple choice before solving
> - Memorize formulas — the ASVAB does not provide a formula sheet