Lecture Notes on Exponent Rules and Algebra Reviews

Exponent Rules

Basics

• Expressions like 2^5 or x^n:
  • 2^5 = 2 * 2 * 2 * 2 * 2
  • x^n = x multiplied by itself n times
  • Base: number being multiplied by itself
  • Exponent (or power): number of times the base is multiplied

Rules

1. Product Rule:

   • (x^n) * (x^m) = x^(n+m)
   • Example: 2^3 * 2^4 = 2^7

2. Quotient Rule:

   • (x^n) / (x^m) = x^(n-m)
   • Example: 3^6 / 3^2 = 3^4

3. Power Rule:

   • (x^n)^m = x^(n*m)
   • Example: (5^4)^3 = 5^(4*3) = 5^12

4. Zero Exponent Rule:

   • x^0 = 1
   • Example: 2^0 = 1

5. Negative Exponent Rule:

   • x^-n = 1 / x^n
   • Example: 5^-7 = 1 / 5^7

6. Fractional Exponents:

   • x^(1/n) = nth root of x
   • Example: 64^(1/3) = 4

7. Exponent Distribution Over Multiplication:

   • (xy)^n = (x^n) * (y^n)
   • Example: (5 * 7)^3 = 5^3 * 7^3

8. Exponent Distribution Over Division:

   • (x/y)^n = (x^n) / (y^n)
   • Example: (2/7)^5 = 2^5 / 7^5*

Important Notes

• Cannot distribute exponents over addition/subtraction.
• Example of incorrect distribution:
  • (a + b)^n ≠ a^n + b^n
  • (2 + 3)^2 ≠ 2^2 + 3^2

Quadratic Equations

Basics

• Standard form: ax^2 + bx + c = 0
• Solutions via:
  • Factoring
  • Quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a

Graphing Quadratics

• Parabola shape
• Vertex: Lowest point (if opens up) or highest point (if opens down)
• Axis of symmetry: line through the vertex
• Finding vertex in standard form using vertex formula: x = -b / 2a

Example Problems

• Solving by factoring and grouping
• Solving using quadratic formula
• Simplifying expressions using exponent rules

Rational Expressions

Simplification

• Factor numerator and denominator
• Cancel common factors

Multiplication/Division

• Multiply: (a/b) * (c/d) = (a*c) / (b*d)
• Divide: (a/b) / (c/d) = (a/b) * (d/c)

Addition/Subtraction

• Find common denominator
• Combine numerators
• Simplify

Examples

• Simplifying complex rational expressions
• Multiplying and dividing rational expressions
• Adding and subtracting rational expressions

Radical Expressions

Basics

• Radical sign indicates root
• Square root: sqrt(x)
• Cube root: ^(1/3)(x)

Simplifying Radicals

• Distribute radical sign across multiplication/division
• Example: sqrt(9 * 16) = sqrt(9) * sqrt(16)
• Cannot distribute radicals over addition/subtraction

Rationalizing Denominators

• Eliminate radicals from denominator by multiplying numerator and denominator by an appropriate radical
• Example: 1 / sqrt(x) * sqrt(x) / sqrt(x) = sqrt(x) / x*

Examples

• Various simplifications of radical expressions
• Rationalizing denominators in complex fractions

Factoring Techniques

Basics

• Writing expressions as product of factors

Methods

1. Greatest Common Factor (GCF)

   • Example: 15x + 25 = 5(3x + 5)

2. Factoring by Grouping

   • Example: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

3. Factoring Quadratics

   • Example: x^2 + 5x + 6 = (x + 2)(x + 3)

4. Difference of Squares

   • Example: a^2 - b^2 = (a + b)(a - b)

5. Sum/Difference of Cubes

   • Example: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
   • Example: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Advanced Problems

• Combining multiple techniques to factor complex expressions

Solving Equations

Linear Equations

• Standard form: ax + b = 0
• Solve for x

Systems of Equations

• Methods:
  • Substitution
  • Elimination

Quadratic Equations

• Use of quadratic formula
• Completing the square

Rational Equations

• Clear denominators
• Solve resulting polynomial equations

Radical Equations

• Isolate radical
• Square both sides
• Solve resulting polynomial equation
• Check for extraneous solutions

Absolute Value Equations

• Isolate absolute value
• Set up two equations
• Solve each

Examples

• Solving various equations using above methods

Additional Topics

Exponential and Logarithmic Functions

• Exponential functions: f(x) = a * b^x
• Logarithmic functions: log_b(x) = y <-> b^y = x
• Properties and rules of logarithms
• Solving exponential/logarithmic equations*

Graphing Functions

• Basic toolkit functions
• Transformations (shifts, stretches, reflections)
• Combining functions (addition, subtraction, multiplication, division)
• Inverse functions and their properties

Application Problems

• Distance, rate, and time problems
• Mixture problems involving solutions
• Population growth and decay modeling

Key Concepts

• Understanding and applying the properties of exponents and logarithms
• Factoring techniques and solving various types of equations
• Graphing and transforming functions
• Practical application of algebra in real-world problems