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.
Laws of Exponents
Formulas
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
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
(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
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
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
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
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
Base
Example
In 3⁴, the base is 3
The number being multiplied repeatedly in exponential form
Exponent/Power/Index
Example
In 3⁴, the exponent is 4
The number indicating how many times the base is multiplied by itself
Scientific Notation
Example
5,000 = 5 × 10³
Writing numbers as a × 10ⁿ where 1 ≤ a < 10
Basic Algebra Operations
Formulas
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
(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
(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
(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
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
Variable
Example
In 2x + 3 = 7, x is the variable
A letter representing an unknown number
Coefficient
Example
In 5x, the coefficient is 5
The number multiplied by a variable
Like terms
Example
3x and -7x are like terms
Terms with same variable and same power
Expression
Example
3x + 2y - 5
Mathematical phrase with numbers, variables, and operations
Reactions or equations
a² - b² = (a + b)(a - b)
Note
Only works for subtraction, not addition
Conditions
Difference of squares factoring
Number Series Types
Formulas
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
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
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
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
Arithmetic Sequence
Example
2, 5, 8, 11, ... (common difference = 3)
Sequence where difference between consecutive terms is constant
Geometric Sequence
Example
3, 6, 12, 24, ... (common ratio = 2)
Sequence where ratio between consecutive terms is constant
Common Difference (d)
Example
In 1, 4, 7, 10, ..., d = 3
Constant added to each term in arithmetic sequence
Common Ratio (r)
Example
In 2, 6, 18, 54, ..., r = 3
Constant multiplied to each term in geometric sequence
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
PEMDAS
Example
2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50
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
Arithmetic vs Geometric Sequences
| Characteristic | Arithmetic | Geometric |
|---|---|---|
| Pattern Type | Add constant difference | Multiply by constant ratio |
| Formula | aₙ = a₁ + (n-1)d | aₙ = a₁ × r^(n-1) |
| Example | 2, 5, 8, 11, 14 | 2, 6, 18, 54, 162 |
| Key Value | Common difference (d) | Common ratio (r) |
Positive vs Negative Exponents
| Exponent Type | Rule | Example | Result |
|---|---|---|---|
| Positive | aⁿ = a × a × ... (n times) | 2³ | 8 |
| Negative | a⁻ⁿ = 1/aⁿ | 2⁻³ | 1/8 |
| Zero | a⁰ = 1 (a ≠ 0) | 5⁰ | 1 |
Diagrams
Algebra, Exponents & Number Series Concept Map
Complete overview of all major topics in the chapter showing relationships between concepts
Exponent Rules Decision Tree
Step-by-step decision process for applying exponent rules correctly
Number Series Pattern Recognition
Systematic approach to identify and continue number sequence patterns