Civil Service Exam (Subprofessional) Numerical Ability — Algebra, Exponents & Number SeriesCheat Sheet
Algebra, Exponents & Number Series cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Algebra, Exponents & Number Series for Civil Service Exam (Subprofessional) Numerical Ability. Download, print, revise.
Exam context
The Career Service Examination — Subprofessional Level is conducted by Civil Service Commission (CSC) and is scheduled for Bi-annual — March and August 2026. The Numerical Ability subtest is marked as "~25% weightage" in the official pattern, and Algebra, Exponents & Number Series appears in position 5th of 9 in the Civil Service Exam (Subprofessional) Numerical Ability review rotation. Passing mark: 80%. Recent Civil Service Exam (Subprofessional) 2026 papers have drawn roughly 17 questions from this subject.
Algebra, Exponents & Number Series - Cheat sheet
Your last-minute revision companion for mastering algebraic operations, exponent rules, and number pattern recognition for Philippine entrance exams.
Sections
Formulas
Formula
a^m × a^n = a^(m+n)
Meaning
Product Rule - keep base, add exponents when multiplying same bases
Watch Out
Only works with SAME bases - cannot combine 2³ × 3² this way
When To Use
When multiplying expressions with same base but different exponents
Formula
a^m ÷ a^n = a^(m-n)
Meaning
Quotient Rule - keep base, subtract exponents when dividing same bases
Watch Out
Remember: subtract the BOTTOM exponent from the TOP exponent
When To Use
When dividing expressions with same base
Formula
(a^m)^n = a^(mn)
Meaning
Power Rule - multiply exponents when raising a power to another power
Watch Out
Don't confuse with addition - you MULTIPLY the exponents, not add them
When To Use
When you see parentheses around a base with exponent, raised to another power
Formula
a^0 = 1
Meaning
Zero Exponent Rule - anything (except 0) raised to power 0 equals 1
Watch Out
This rule does NOT apply to 0⁰ which is undefined
When To Use
When you see any base with exponent 0
Formula
a^(-m) = 1/a^m
Meaning
Negative Exponent Rule - negative exponent means reciprocal with positive exponent
Watch Out
The base doesn't become negative - only the position changes (numerator ↔ denominator)
When To Use
When dealing with negative exponents
Formula
a^(1/n) = ⁿ√a
Meaning
Fractional Exponent Rule - 1/n exponent means nth root
Watch Out
Don't forget: a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m
When To Use
When exponent is a fraction
Common Values
Value
1², 2², 3², 4², 5² = 1, 4, 9, 16, 25
Symbol
n²
Quantity
Perfect squares
Value
1³, 2³, 3³, 4³, 5³ = 1, 8, 27, 64, 125
Symbol
n³
Quantity
Perfect cubes
Value
2¹, 2², 2³, 2⁴, 2⁵ = 2, 4, 8, 16, 32
Symbol
2ⁿ
Quantity
Powers of 2
Section Title
Laws of Exponents
Important Facts
- Product rule only applies to same bases: 2³ × 2⁵ = 2⁸, but 2³ × 3⁵ cannot be simplified
- When no exponent is shown, it's understood to be 1
- Negative exponents don't make the answer negative - they create fractions
- Zero raised to any positive power is zero: 0ⁿ = 0 (n > 0)
- One raised to any power is one: 1ⁿ = 1
Key Definitions
Term
Base
Example
In 3⁴, the base is 3
Definition
The number being multiplied repeatedly in exponential form
Term
Exponent/Power/Index
Example
In 3⁴, the exponent is 4
Definition
The number indicating how many times the base is multiplied by itself
Term
Scientific Notation
Example
5,000 = 5 × 10³
Definition
Writing numbers as a × 10ⁿ where 1 ≤ a < 10
Formulas
Formula
ax + b = c → x = (c - b)/a
Meaning
Solving linear equations - isolate variable on one side
Watch Out
Always perform same operation on both sides of the equation
When To Use
For simple one-variable linear equations
Formula
(a + b)(c + d) = ac + ad + bc + bd
Meaning
FOIL method - distribute each term in first bracket to each term in second
Watch Out
Don't forget to combine like terms after expanding
When To Use
When multiplying two binomials
Formula
(a + b)² = a² + 2ab + b²
Meaning
Perfect square expansion - square of sum
Watch Out
Don't forget the middle term 2ab - (a + b)² ≠ a² + b²
When To Use
When squaring a binomial sum
Formula
(a - b)² = a² - 2ab + b²
Meaning
Perfect square expansion - square of difference
Watch Out
Middle term is negative: -2ab, not +2ab
When To Use
When squaring a binomial difference
Section Title
Basic Algebra Operations
Important Facts
- Combine only like terms: 3x + 2y cannot be simplified further
- When multiplying by negative, inequality sign flips: -2x > 6 becomes x < -3
- Distributive property: a(b + c) = ab + ac
- Order of operations: PEMDAS (Parentheses, Exponents, Multiply/Divide, Add/Subtract)
- Zero product property: if ab = 0, then a = 0 or b = 0
Key Definitions
Term
Variable
Example
In 2x + 3 = 7, x is the variable
Definition
A letter representing an unknown number
Term
Coefficient
Example
In 5x, the coefficient is 5
Definition
The number multiplied by a variable
Term
Like terms
Example
3x and -7x are like terms
Definition
Terms with same variable and same power
Term
Expression
Example
3x + 2y - 5
Definition
Mathematical phrase with numbers, variables, and operations
Reactions Or Equations
Note
Only works for subtraction, not addition
Equation
a² - b² = (a + b)(a - b)
Conditions
Difference of squares factoring
Formulas
Formula
aₙ = a₁ + (n-1)d
Meaning
Arithmetic sequence nth term formula: first term + (position-1) × common difference
Watch Out
Don't forget to subtract 1 from n: it's (n-1)d, not nd
When To Use
Finding any term in arithmetic sequence
Formula
aₙ = a₁ × r^(n-1)
Meaning
Geometric sequence nth term formula: first term × common ratio^(position-1)
Watch Out
Exponent is (n-1), not n
When To Use
Finding any term in geometric sequence
Formula
Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂
Meaning
Each term equals sum of two preceding terms
Watch Out
Need at least first two terms to generate sequence
When To Use
When each term is sum of previous two terms
Section Title
Number Series Types
Important Facts
- Arithmetic: constant DIFFERENCE (add/subtract same amount)
- Geometric: constant RATIO (multiply/divide by same amount)
- Fibonacci: each term = sum of previous two terms
- Perfect squares: 1, 4, 9, 16, 25, ... (n²)
- Perfect cubes: 1, 8, 27, 64, 125, ... (n³)
- Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
- Two-stage sequences: look at differences between terms for pattern
Key Definitions
Term
Arithmetic Sequence
Example
2, 5, 8, 11, ... (common difference = 3)
Definition
Sequence where difference between consecutive terms is constant
Term
Geometric Sequence
Example
3, 6, 12, 24, ... (common ratio = 2)
Definition
Sequence where ratio between consecutive terms is constant
Term
Common Difference (d)
Example
In 1, 4, 7, 10, ..., d = 3
Definition
Constant added to each term in arithmetic sequence
Term
Common Ratio (r)
Example
In 2, 6, 18, 54, ..., r = 3
Definition
Constant multiplied to each term in geometric sequence
Section Title
Problem-Solving Steps
Important Facts
- Always simplify inside parentheses first
- Exponents come before multiplication and division
- Multiplication and division are equal priority (left to right)
- Addition and subtraction are equal priority (left to right)
- When solving equations, perform same operation on both sides
- Check answers by substituting back into original equation
Key Definitions
Term
PEMDAS
Example
2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50
Definition
Order of operations: Parentheses, Exponents, Multiply/Divide, Add/Subtract
Must Remember
- Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ (same base only)
- Negative exponent: a⁻ⁿ = 1/aⁿ (creates fraction, not negative number)
- Zero exponent: a⁰ = 1 (except 0⁰ which is undefined)
- Perfect square expansion: (a + b)² = a² + 2ab + b² (don't forget 2ab term)
- Arithmetic sequence: constant difference between terms
- Geometric sequence: constant ratio between terms
- PEMDAS order: Parentheses → Exponents → Multiply/Divide → Add/Subtract
- Fibonacci: each term = sum of previous two terms
- When multiplying negative inequality: flip the inequality sign
- Always check your answer by substituting back into original equation
Last Minute Tips
- For number series: first check if arithmetic (constant difference), then geometric (constant ratio), then look for squares/cubes
- With exponents: write out the first few steps if unsure - don't rely on mental math for complex expressions
- For algebra: isolate the variable step by step - don't try to do multiple operations at once
- In word problems: identify what you're solving for first, then translate words to mathematical expressions
- Double-check negative signs - they're the #1 source of careless errors in algebra and exponents
Comparison Tables
Rows
Values
- Add constant difference
- Multiply by constant ratio
Property
Pattern Type
Values
- aₙ = a₁ + (n-1)d
- aₙ = a₁ × r^(n-1)
Property
Formula
Values
- 2, 5, 8, 11, 14
- 2, 6, 18, 54, 162
Property
Example
Values
- Common difference (d)
- Common ratio (r)
Property
Key Value
Columns
- Characteristic
- Arithmetic
- Geometric
Table Title
Arithmetic vs Geometric Sequences
Rows
Values
- aⁿ = a × a × ... (n times)
- 2³
- 8
Property
Positive
Values
- a⁻ⁿ = 1/aⁿ
- 2⁻³
- 1/8
Property
Negative
Values
- a⁰ = 1 (a ≠ 0)
- 5⁰
- 1
Property
Zero
Columns
- Exponent Type
- Rule
- Example
- Result
Table Title
Positive vs Negative Exponents
Previous chapter
Ratio, Proportion & Percentage
Next chapter
Word Problems — Speed/Distance/Age, Discount & Interest
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