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Exponent Rules Overview

Sep 16, 2025

Overview

This lecture covers the basic properties and rules of exponents (powers) when the exponents are positive whole numbers, using simplified examples and step-by-step explanations.

Definition and Notation of Exponents

  • An exponent indicates how many times a number (the base) is multiplied by itself.
  • Example: (3^5 = 3 \times 3 \times 3 \times 3 \times 3).

Basic Exponent Rules

  • Product Rule: (a^m \times a^n = a^{m+n}).
    • Example: (3^4 \times 3^2 = 3^6).
  • Quotient Rule: (\frac{a^m}{a^n} = a^{m-n}).
    • Example: (\frac{3^4}{3^2} = 3^{4-2} = 3^2).
  • Power of a Power Rule: ((a^m)^n = a^{m \times n}).
    • Example: ((2^3)^2 = 2^{3 \times 2} = 2^6).
  • Power of a Product Rule: ((ab)^m = a^m \times b^m).
    • Example: ((2x)^3 = 2^3 \times x^3 = 8x^3).
  • Power of a Quotient Rule: (\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}).
    • Example: (\left(\frac{2}{x}\right)^3 = \frac{2^3}{x^3} = \frac{8}{x^3}).

Examples Using Multiple Rules

  • ((2^4)^3 = 2^{4 \times 3} = 2^{12}).
  • ((2x)^3 = 8x^3).
  • (\frac{(x^4)^3}{x^2} = \frac{x^{4 \times 3}}{x^2} = \frac{x^{12}}{x^2} = x^{10}).
  • (\frac{y^5)^3 \times (y^3)^2}{(y^4)^4} = \frac{y^{15} \times y^6}{y^{16}} = \frac{y^{21}}{y^{16}} = y^5).
  • (\frac{x^2 y^3}{x^4}) all raised to the 5th power simplifies as (\frac{y^{15}}{x^{10}}).

Key Terms & Definitions

  • Exponent — The number that indicates how many times to multiply the base by itself.
  • Base — The number being multiplied in an exponential expression.
  • Product Rule — When multiplying exponents with the same base, add the exponents.
  • Quotient Rule — When dividing exponents with the same base, subtract the exponents.
  • Power of a Power — Raise a power to another power by multiplying the exponents.
  • Power of a Product — Distribute the exponent to each factor inside the parentheses.
  • Power of a Quotient — Distribute the exponent to both numerator and denominator.

Action Items / Next Steps

  • Practice simplifying expressions using these exponent rules.
  • Review additional examples involving negative and fractional exponents in future lessons.

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