MSU-SASE Mathematics — Algebra — Sets, Exponents, Radicals, Polynomials & EquationsCheat Sheet
Algebra — Sets, Exponents, Radicals, Polynomials & Equations cheat sheet for MSU-SASE aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. Mindanao State University's most-tested concepts, all in one place.
Exam context
On the MSU-SASE 2026, the Mathematics subtest carries a "Core" weight in Mindanao State University's pattern. Algebra — Sets, Exponents, Radicals, Polynomials & Equations lands at position 3rd out of 9 in the standard review order. Target score is Competitive overall score, and roughly a meaningful share of items come from Mathematics on a typical MSU-SASE paper.
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
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