Algebra — Sets, Exponents, Radicals, Polynomials & Equations - Cheat sheet
Your last-minute revision companion for mastering algebra fundamentals. Every formula, rule, and concept you need for UPCAT success.
Set Theory Fundamentals
Formulas
|A ∪ B| = |A| + |B| - |A ∩ B|
Meaning
|A| = cardinality of set A, |B| = cardinality of set B, |A ∩ B| = cardinality of intersection
Watch out
Don't forget to subtract intersection to avoid double counting
When to use
Finding number of elements in union of two sets
|A'| = |U| - |A|
Meaning
|A'| = cardinality of complement of A, |U| = cardinality of universal set
Watch out
Make sure you know what the universal set U contains
When to use
Finding elements NOT in set A
Important facts
- Empty set ∅ is subset of every set
- A ∪ A' = U (universal set)
- A ∩ A' = ∅ (empty set)
- Number of subsets of set with n elements = 2ⁿ
- Venn diagrams help visualize set operations
Key definitions
Set
Example
A = {1, 2, 3, 4}
Well-defined collection of distinct objects
Subset
Example
{1, 2} ⊆ {1, 2, 3}
Set A is subset of B if every element of A is in B (A ⊆ B)
Universal Set
Example
For integers: U = Z
Set containing all elements under consideration
Complement
Example
If U = {1,2,3,4} and A = {1,2}, then A' = {3,4}
Elements in universal set but not in given set (A')
Diagrams to know
- Venn diagrams for 2 and 3 sets
- Set notation symbols
- Universal set representation
Reactions or equations
A ∪ B = B ∪ A (Commutative)
Note
Order doesn't matter in union
Conditions
For any sets A and B
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Note
Intersection distributes over union
Conditions
Distributive law
Laws of Exponents
Formulas
aᵐ × aⁿ = aᵐ⁺ⁿ
Meaning
a = base, m and n = exponents
Watch out
Bases must be identical - can't use with different bases
When to use
Multiplying powers with same base
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Meaning
a ≠ 0, m and n = exponents
Watch out
Subtract exponents, don't divide them
When to use
Dividing powers with same base
(aᵐ)ⁿ = aᵐⁿ
Meaning
a = base, m and n = exponents
Watch out
Multiply exponents, don't add them
When to use
Taking power of a power
a⁻ⁿ = 1/aⁿ
Meaning
a ≠ 0, n = positive integer
Watch out
Negative exponent creates reciprocal, not negative number
When to use
Converting negative exponents
a⁰ = 1
Meaning
a ≠ 0
Watch out
0⁰ is undefined
When to use
Any non-zero number to power zero
Common values
Value
1,024
Symbol
2¹⁰
Quantity
2¹⁰
Value
1,000
Symbol
10³
Quantity
10³
Important facts
- Product rule: add exponents when multiplying same bases
- Quotient rule: subtract exponents when dividing same bases
- Power rule: multiply exponents when raising power to power
- Negative exponents create reciprocals
- Zero exponent always equals 1 (except 0⁰)
Key definitions
Base
Example
In 5³, base is 5
Number being multiplied by itself
Exponent
Example
In 5³, exponent is 3
Number indicating how many times base is used as factor
Scientific Notation
Example
3,000 = 3 × 10³
Number written as a × 10ⁿ where 1 ≤ a < 10
Diagrams to know
- Exponent notation breakdown
- Scientific notation place value chart
Reactions or equations
(ab)ⁿ = aⁿbⁿ
Note
Power of product equals product of powers
Conditions
For any real numbers a, b and integer n
(a/b)ⁿ = aⁿ/bⁿ
Note
Power of quotient equals quotient of powers
Conditions
b ≠ 0
Radicals and Roots
Formulas
√(ab) = √a × √b
Meaning
a ≥ 0, b ≥ 0 for square roots
Watch out
Only works for multiplication, NOT addition: √(a+b) ≠ √a + √b
When to use
Simplifying radical products
√(a/b) = √a / √b
Meaning
a ≥ 0, b > 0 for square roots
Watch out
Denominator cannot be zero
When to use
Simplifying radical quotients
ⁿ√aᵐ = a^(m/n)
Meaning
n = index of root, m = exponent of radicand
Watch out
Fractional exponent: numerator is power, denominator is root
When to use
Converting between radical and exponential form
(√a)² = a
Meaning
a ≥ 0
Watch out
Result is always non-negative
When to use
Squaring a square root
Common values
Value
≈ 1.414
Symbol
√2
Quantity
√2
Value
≈ 1.732
Symbol
√3
Quantity
√3
Value
≈ 2.236
Symbol
√5
Quantity
√5
Important facts
- √a² = |a| (absolute value for even roots)
- Cube roots can be negative: ³√(-8) = -2
- Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
- Perfect cubes: 1, 8, 27, 64, 125
- Cannot take even root of negative number in real numbers
Key definitions
Radical
Example
√25 = 5
Expression containing root symbol (√)
Radicand
Example
In √16, radicand is 16
Number under the radical sign
Index
Example
In ³√8, index is 3
Number indicating which root to take
Rationalize
Example
1/√2 = √2/2
Eliminate radicals from denominator
Diagrams to know
- Radical notation parts (index, radicand, radical sign)
- Perfect square and cube charts
Reactions or equations
√a × √a = a
Note
Square root times itself equals the radicand
Conditions
a ≥ 0
ⁿ√a^n = a if n is odd, |a| if n is even
Note
Even roots always give non-negative results
Conditions
For real numbers
Polynomial Operations
Formulas
(a + b)² = a² + 2ab + b²
Meaning
a, b = any real numbers or expressions
Watch out
Don't forget the middle term 2ab
When to use
Squaring a binomial sum
(a - b)² = a² - 2ab + b²
Meaning
a, b = any real numbers or expressions
Watch out
Middle term is negative: -2ab
When to use
Squaring a binomial difference
(a + b)(a - b) = a² - b²
Meaning
a, b = any real numbers or expressions
Watch out
Results in difference of squares only
When to use
Multiplying sum and difference of same terms
a³ + b³ = (a + b)(a² - ab + b²)
Meaning
Sum of cubes factorization
Watch out
Second factor has minus sign in middle
When to use
Factoring sum of perfect cubes
a³ - b³ = (a - b)(a² + ab + b²)
Meaning
Difference of cubes factorization
Watch out
Second factor has plus sign in middle
When to use
Factoring difference of perfect cubes
Important facts
- Add/subtract polynomials by combining like terms
- Multiply polynomials by distributing each term
- FOIL method: First, Outer, Inner, Last for binomials
- Degree of product = sum of degrees
- Always arrange in descending order of powers
Key definitions
Polynomial
Example
3x² + 2x - 5
Expression with variables and coefficients using addition, subtraction, multiplication
Degree
Example
Degree of 4x³ + 2x - 1 is 3
Highest power of variable in polynomial
Leading Coefficient
Example
In 5x³ + 2x² + 1, leading coefficient is 5
Coefficient of term with highest degree
Like Terms
Example
3x² and -7x² are like terms
Terms with same variables and same exponents
Diagrams to know
- FOIL method diagram
- Polynomial addition/subtraction alignment
- Factoring tree diagrams
Reactions or equations
(ax + b)(cx + d) = acx² + (ad + bc)x + bd
Note
Use FOIL: First, Outer, Inner, Last
Conditions
General form of binomial multiplication
Quadratic Equations
Formulas
x = (-b ± √(b² - 4ac)) / (2a)
Meaning
For ax² + bx + c = 0, where a ≠ 0
Watch out
Check discriminant b² - 4ac for number of real solutions
When to use
Solving any quadratic equation
x² = k → x = ±√k
Meaning
k ≥ 0 for real solutions
Watch out
Don't forget the ± sign
When to use
Solving by taking square root
Discriminant = b² - 4ac
Meaning
Determines nature of roots
Watch out
0: two real, = 0: one real, < 0: no real solutions
When to use
Finding number and type of solutions
Important facts
- Standard form: ax² + bx + c = 0
- Zero Product Property: if ab = 0, then a = 0 or b = 0
- Graph is a parabola
- Vertex form: y = a(x - h)² + k
- Sum of roots = -b/a, Product of roots = c/a
Key definitions
Quadratic Equation
Example
2x² - 5x + 3 = 0
Equation of form ax² + bx + c = 0 where a ≠ 0
Discriminant
Example
For x² - 4x + 4 = 0, discriminant = 0
Expression b² - 4ac that determines nature of roots
Completing the Square
Example
x² + 6x + 9 = (x + 3)²
Method to solve quadratic by making perfect square trinomial
Diagrams to know
- Parabola graph showing vertex, axis of symmetry
- Quadratic formula derivation steps
- Discriminant decision tree
Reactions or equations
x² + 2hx + h² = (x + h)²
Note
Used in completing the square method
Conditions
Perfect square trinomial
Linear Equations and Systems
Formulas
y = mx + b
Meaning
m = slope, b = y-intercept
Watch out
Slope is rise over run: Δy/Δx
When to use
Slope-intercept form of line
m = (y₂ - y₁)/(x₂ - x₁)
Meaning
Slope between two points (x₁,y₁) and (x₂,y₂)
Watch out
Make sure x₂ ≠ x₁ to avoid division by zero
When to use
Finding slope from two points
y - y₁ = m(x - x₁)
Meaning
Point-slope form with point (x₁,y₁) and slope m
Watch out
Substitute the known values carefully
When to use
Writing equation when you know point and slope
Important facts
- Elimination method: add/subtract equations to eliminate variable
- Substitution method: solve one equation for variable, substitute
- Parallel lines: same slope, different y-intercepts
- Perpendicular lines: slopes are negative reciprocals
- Horizontal line: slope = 0, Vertical line: undefined slope
Key definitions
Linear Equation
Example
3x + 2y = 6
Equation whose graph is a straight line (degree 1)
System of Equations
Example
2x + y = 5 and x - y = 1
Set of equations solved simultaneously
Parallel Lines
Example
y = 2x + 1 and y = 2x + 5
Lines with same slope, never intersect
Perpendicular Lines
Example
y = 2x and y = -½x
Lines whose slopes multiply to -1
Diagrams to know
- Coordinate plane with quadrants
- Slope visualization (rise over run)
- System of equations solution graphs
Reactions or equations
Ax + By = C
Note
A and B cannot both be zero
Conditions
Standard form of linear equation
Must remember
- Quadratic Formula: x = (-b ± √(b² - 4ac))/(2a)
- Perfect Square: (a ± b)² = a² ± 2ab + b²
- Difference of Squares: a² - b² = (a + b)(a - b)
- Exponent Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- Zero Exponent: a⁰ = 1 (a ≠ 0)
- Negative Exponent: a⁻ⁿ = 1/aⁿ
- Set Cardinality: |A ∪ B| = |A| + |B| - |A ∩ B|
- Radical Product: √(ab) = √a × √b
- Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)
- FOIL Method: (a + b)(c + d) = ac + ad + bc + bd
Last minute tips
- In set problems, draw Venn diagrams to visualize - prevents counting errors
- For quadratic equations, always check discriminant first to know how many real solutions exist
- When simplifying radicals, look for perfect square factors first
- Remember negative exponents create reciprocals, not negative numbers
- In system of equations, substitution works best when one variable has coefficient 1
Comparison tables
Quadratic Solution Methods
| Method | When to Use | Pros | Cons |
|---|---|---|---|
| Factoring | When polynomial factors nicely | Quick and simple | Only works for factorable equations |
| Square Root | When in form x² = k | Very fast | Limited to specific forms |
| Completing Square | Any quadratic | Shows vertex form | More steps involved |
| Quadratic Formula | Any quadratic | Always works | Requires memorization |
Exponent Rules Summary
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ |
| Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁴ ÷ 5² = 5² |
| Power Rule | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| Zero Exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
Diagrams
Algebra Core Concepts Overview
Complete overview of all major algebra topics covered in this chapter
Quadratic Equation Solution Strategy
Decision tree for choosing the best method to solve quadratic equations
Exponent Rules Application Guide
Step-by-step guide for applying exponent rules correctly