CEUET Mathematics — Algebra — Sets, Exponents, Radicals, Polynomials & EquationsCheat Sheet
One-page cheat sheet for CEUET Mathematics — Algebra — Sets, Exponents, Radicals, Polynomials & Equations. Every formula, definition, and key fact you need for this chapter, condensed to a single printable page. Designed for the final review session before the CEUET 2026.
Exam context
Centro Escolar University runs the Centro Escolar University Entrance Test on Q3–Q4 2026. Its Mathematics section sits under a "Core" weighting, and Algebra — Sets, Exponents, Radicals, Polynomials & Equations is the 3rd chapter in the 9-chapter CEUET Mathematics rotation. The CEUET passing mark is Competitive overall score, and the most recent 2026 paper drew about a meaningful share of questions from Mathematics.
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.
Sections
Formulas
Formula
|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
Formula
|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
Section Title
Set Theory Fundamentals
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
Term
Set
Example
A = {1, 2, 3, 4}
Definition
Well-defined collection of distinct objects
Term
Subset
Example
{1, 2} ⊆ {1, 2, 3}
Definition
Set A is subset of B if every element of A is in B (A ⊆ B)
Term
Universal Set
Example
For integers: U = Z
Definition
Set containing all elements under consideration
Term
Complement
Example
If U = {1,2,3,4} and A = {1,2}, then A' = {3,4}
Definition
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
Note
Order doesn't matter in union
Equation
A ∪ B = B ∪ A (Commutative)
Conditions
For any sets A and B
Note
Intersection distributes over union
Equation
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Conditions
Distributive law
Formulas
Formula
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
Formula
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
Formula
(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
Formula
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
Formula
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
Section Title
Laws of Exponents
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
Term
Base
Example
In 5³, base is 5
Definition
Number being multiplied by itself
Term
Exponent
Example
In 5³, exponent is 3
Definition
Number indicating how many times base is used as factor
Term
Scientific Notation
Example
3,000 = 3 × 10³
Definition
Number written as a × 10ⁿ where 1 ≤ a < 10
Diagrams To Know
- Exponent notation breakdown
- Scientific notation place value chart
Reactions Or Equations
Note
Power of product equals product of powers
Equation
(ab)ⁿ = aⁿbⁿ
Conditions
For any real numbers a, b and integer n
Note
Power of quotient equals quotient of powers
Equation
(a/b)ⁿ = aⁿ/bⁿ
Conditions
b ≠ 0
Formulas
Formula
√(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
Formula
√(a/b) = √a / √b
Meaning
a ≥ 0, b > 0 for square roots
Watch Out
Denominator cannot be zero
When To Use
Simplifying radical quotients
Formula
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
Formula
(√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
Section Title
Radicals and Roots
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
Term
Radical
Example
√25 = 5
Definition
Expression containing root symbol (√)
Term
Radicand
Example
In √16, radicand is 16
Definition
Number under the radical sign
Term
Index
Example
In ³√8, index is 3
Definition
Number indicating which root to take
Term
Rationalize
Example
1/√2 = √2/2
Definition
Eliminate radicals from denominator
Diagrams To Know
- Radical notation parts (index, radicand, radical sign)
- Perfect square and cube charts
Reactions Or Equations
Note
Square root times itself equals the radicand
Equation
√a × √a = a
Conditions
a ≥ 0
Note
Even roots always give non-negative results
Equation
ⁿ√a^n = a if n is odd, |a| if n is even
Conditions
For real numbers
Formulas
Formula
(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
Formula
(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
Formula
(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
Formula
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
Formula
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
Section Title
Polynomial Operations
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
Term
Polynomial
Example
3x² + 2x - 5
Definition
Expression with variables and coefficients using addition, subtraction, multiplication
Term
Degree
Example
Degree of 4x³ + 2x - 1 is 3
Definition
Highest power of variable in polynomial
Term
Leading Coefficient
Example
In 5x³ + 2x² + 1, leading coefficient is 5
Definition
Coefficient of term with highest degree
Term
Like Terms
Example
3x² and -7x² are like terms
Definition
Terms with same variables and same exponents
Diagrams To Know
- FOIL method diagram
- Polynomial addition/subtraction alignment
- Factoring tree diagrams
Reactions Or Equations
Note
Use FOIL: First, Outer, Inner, Last
Equation
(ax + b)(cx + d) = acx² + (ad + bc)x + bd
Conditions
General form of binomial multiplication
Formulas
Formula
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
Formula
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
Formula
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
Section Title
Quadratic Equations
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
Term
Quadratic Equation
Example
2x² - 5x + 3 = 0
Definition
Equation of form ax² + bx + c = 0 where a ≠ 0
Term
Discriminant
Example
For x² - 4x + 4 = 0, discriminant = 0
Definition
Expression b² - 4ac that determines nature of roots
Term
Completing the Square
Example
x² + 6x + 9 = (x + 3)²
Definition
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
Note
Used in completing the square method
Equation
x² + 2hx + h² = (x + h)²
Conditions
Perfect square trinomial
Formulas
Formula
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
Formula
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
Formula
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
Section Title
Linear Equations and Systems
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
Term
Linear Equation
Example
3x + 2y = 6
Definition
Equation whose graph is a straight line (degree 1)
Term
System of Equations
Example
2x + y = 5 and x - y = 1
Definition
Set of equations solved simultaneously
Term
Parallel Lines
Example
y = 2x + 1 and y = 2x + 5
Definition
Lines with same slope, never intersect
Term
Perpendicular Lines
Example
y = 2x and y = -½x
Definition
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
Note
A and B cannot both be zero
Equation
Ax + By = C
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
Rows
Values
- When polynomial factors nicely
- Quick and simple
- Only works for factorable equations
Property
Factoring
Values
- When in form x² = k
- Very fast
- Limited to specific forms
Property
Square Root
Values
- Any quadratic
- Shows vertex form
- More steps involved
Property
Completing Square
Values
- Any quadratic
- Always works
- Requires memorization
Property
Quadratic Formula
Columns
- Method
- When to Use
- Pros
- Cons
Table Title
Quadratic Solution Methods
Rows
Values
- aᵐ × aⁿ = aᵐ⁺ⁿ
- 2³ × 2² = 2⁵
Property
Product Rule
Values
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- 5⁴ ÷ 5² = 5²
Property
Quotient Rule
Values
- (aᵐ)ⁿ = aᵐⁿ
- (3²)³ = 3⁶
Property
Power Rule
Values
- a⁰ = 1
- 7⁰ = 1
Property
Zero Exponent
Values
- a⁻ⁿ = 1/aⁿ
- 2⁻³ = 1/8
Property
Negative Exponent
Columns
- Rule
- Formula
- Example
Table Title
Exponent Rules Summary
Previous chapter
Ratio & Proportion
Next chapter
Word Problems — Number, Age, Work, Motion, Mixture, Investment
Ready to practise for the CEUET 2026?
Super Tutor's AI review plan adapts to your weak areas and builds a weekly practice schedule around your target CEUET exam date.