Lecture Notes on Exponent Rules and Algebraic Concepts

Basics of Exponents

• Exponential Expressions: Expressions like $a^n$ represent $a$ multiplied by itself $n$ times.
  • Base: Number being multiplied (bottom number).
  • Exponent/Power: Number of times the base is multiplied (top number).

Exponent Rules

1. Product Rule: $x^n \times x^m = x^{n+m}$.
   • Example: $2^3 \times 2^4 = 2^{3+4} = 2^7$.
2. Quotient Rule: $x^n / x^m = x^{n-m}$.
   • Example: $3^6 / 3^2 = 3^{6-2} = 3^4$.
3. Power Rule: $(x^n)^m = x^{n \cdot m}$.
   • Example: $(5^4)^3 = 5^{4\cdot3} = 5^{12}$.
4. Zero Exponent Rule: $x^0 = 1$ (for $x \neq 0$).
   • Reason: $x^n / x^n = x^{n-n} = x^0 = 1$.
5. Negative Exponent Rule: $x^{-n} = 1/x^n$.
   • Example: $5^{-7} = 1/5^7$.
6. Fractional Exponent Rule: $x^{1/n} = \sqrt[n]{x}$.
   • Example: $64^{1/3} = \sqrt[3]{64} = 4$.
7. Distribution in Product: $(xy)^n = x^n \cdot y^n$.
   • Example: $(5 \times 7)^3 = 5^3 \times 7^3$.
8. Distribution in Quotient: $(x/y)^n = x^n / y^n$.
   • Example: $(2/7)^5 = 2^5 / 7^5$.

Application of Rules

• Simplifying Expressions: Use exponent rules to simplify complex expressions.
• Examples:
  • Simplify $3x^{-2}/x^4$: Use negative and quotient rules.
  • Alternately, use shortcut for passing exponents across fraction bars.

Radical and Rational Exponents

• Radicals vs. Exponents: Radicals can be expressed using fractional exponents.
  • Example: $ ext{nth root of } a = a^{1/n}$.
• Rules for Radicals:
  • Product Rule: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
  • Quotient Rule: $\sqrt{a/b} = \sqrt{a}/\sqrt{b}$.
  • Radicals do not distribute over addition or subtraction.
• Rationalizing Denominators: Convert denominator to non-radical form.

Factoring Techniques

1. Greatest Common Factor (GCF): Factor out the largest common factor.
2. Factoring by Grouping: Useful for four-term polynomials.
3. Factoring Quadratics: Using ac method for quadratics like $ax^2+bx+c$.
4. Special Cases:
   • Difference of Squares: $a^2 - b^2 = (a+b)(a-b)$.
   • Sum/Difference of Cubes: $a^3 \pm b^3$.

Linear and Quadratic Equations

• Solving Quadratics: Using factoring, quadratic formula, or completing the square.
• Linear Equations: Solve using isolation and simplification techniques.

Graphing Techniques

• Understanding Graph Shapes: Parabolas for quadratics, hyperbolas, etc.
• Transformations: Shifts, stretches, and reflections.
• Intercepts and Asymptotes: Identify graph characteristics.

Functions and Their Properties

• Function Operations: Add, subtract, multiply, divide functions.
• Function Composition: $(f \circ g)(x) = f(g(x))$.
• Inverse Functions: Reverse roles of inputs and outputs.

Logarithms

• Logarithmic Functions: $\log_a b = c \Rightarrow a^c = b$.
• Properties of Logs:
  • Product Rule: $\log_a(xy) = \log_a x + \log_a y$.
  • Quotient Rule: $\log_a(x/y) = \log_a x - \log_a y$.
  • Power Rule: $\log_a(x^n) = n \cdot \log_a x$.
• Applications: Solving exponential equations, pH, decibels.

Applications of Exponentials

• Exponential Growth/Decay: Population growth, radioactive decay.
• Compound Interest: Continuous and discrete compounding formulas.

These notes cover the fundamental principles and rules related to exponentiation, radicals, and various algebraic manipulations. Understanding these core concepts is crucial for simplifying expressions, solving equations, and analyzing functions effectively.