Simplifying Exponents Lecture Notes

Basic Exponent Properties

• Multiplication of Exponents:

  • When multiplying by a common base, add the exponents.
  • Example: (x^4 \times x^5 = x^{4+5} = x^9).

• Division of Exponents:

  • When dividing by a common base, subtract the exponents.
  • Example: (\frac{x^7}{x^3} = x^{7-3} = x^4).

• Exponents Raised to a Power:

  • When raising one exponent to another, multiply the exponents.
  • Example: ((x^3)^4 = x^{3\times4} = x^{12}).

Special Exponent Rules

• Zero Exponent Rule:

  • Any number raised to the power of zero is 1.
  • Example: (x^0 = 1), (4^0 = 1).

• Negative Exponents:

  • Move the variable with a negative exponent to the other side of the fraction to make the exponent positive.
  • Example: (x^{-3} = \frac{1}{x^3}), (\frac{1}{x^4} = x^{-4}).

Examples and Exercises

• Multiplication and Addition of Exponents:

  • Simplify: (x^4 \times x^9 = x^{13}).

• Exponentiation of a Product:

  • ((3x^2)^3 = 3^3 \times x^{2\times3} = 27x^6).

• Negative Base Exponents:

  • Differences between (-3^2) and ((-3)^2):
    • (-3^2 = -9)
    • ((-3)^2 = 9)
  • Verify using a calculator.

• Combining Like Terms with Exponents:

  • Simplify: (7x^6 \times 5x^4 = 35x^{10}).

Division and Simplification

• Complex Division:

  • Example: (\frac{24x^7y^3}{8x^4y^{-2}} = 3x^3y^{5}).

• Distributive Property with Exponents:

  • Example: (\frac{35x^3y^5}{63x^4y^7}^2) simplifies to (\frac{25x^{14}}{81y^4}).

Concepts to Remember

• Zero Exponents affect only the base raised to the zero power, not any coefficients.
• Negative Exponents indicate reciprocal relationships.
• Simplify expressions by applying exponent rules and reducing fractions as necessary.

These notes cover the key points from the lecture on simplifying exponents. Practice these examples to strengthen your understanding of exponent rules.