Lecture on Multiplying and Dividing Monomials and Powers

Multiplying Monomials

• Basic Rule: When multiplying two monomials with the same base, add the exponents.
  • Example: (4^2 \times 4^3 = 4^{2+3} = 4^5)
  • Example: (3^7 \times 3^4 = 3^{7+4} = 3^{11})
  • Example: (x^2 \times x^3 = x^{2+3} = x^5)
  • Example: (b^6 \times b^8 = b^{6+8} = b^{14})

Dividing Powers

• Basic Rule: When dividing two numbers with the same base, subtract the exponents.
  • Example: (\frac{7^8}{7^3} = 7^{8-3} = 7^5)
  • Example: (\frac{2^6}{2^3} = 2^{6-3} = 2^3 = 8)
• Negative Exponents: A negative exponent indicates reciprocal.
  • Example: (\frac{x^3}{x^7} = x^{3-7} = x^{-4} = \frac{1}{x^4})

Multiplying Monomials with Different Constants and Variables

• Multiply constants first, then apply rules for exponents.
  • Example: (3x^2 \times 5x^4 = 15x^{2+4} = 15x^6)
  • Example: (6x^2y^3 \times 4x^3y^5 = 24x^{2+3}y^{3+5} = 24x^5y^8)

Complex Examples

• Multiplication and Division: Combined Problems
  • Example: (\frac{y^3 \times y^6}{y^4} = y^{3+6-4} = y^5)
  • Example: (\frac{a^4 \times a^5}{a^3} = a^{4+5-3} = a^{6})
  • Example: (\frac{12x^4y^8}{4x^3y^4} = 3x^{4-3}y^{8-4} = 3xy^4)
  • Example: (\frac{35x^7y^9}{63x^4y^5} = \frac{5}{9}x^{7-4}y^{9-5} = \frac{5x^3y^4}{9})

Special Cases

• Different Bases, Same Exponent: Multiply the bases.
  • Example: (2^5 \times 4^5 = (2 \times 4)^5 = 8^5)
• Converting Bases or Exponents
  • Example: (2^5 \times 4^3 = 2^5 \times (2^2)^3 = 2^5 \times 2^6 = 2^{11})

Algebra Course Overview on Udemy

• Course Content Includes:
  • Basic arithmetic, fractions, solving linear equations, order of operations
  • Graphing linear equations, inequalities, absolute value expressions
  • Polynomials, factoring, systems of equations, quadratic equations
  • Rational and radical expressions, complex numbers, exponential functions, logarithms
  • Functions, conic sections, arithmetic and geometric sequences
• Features:
  • Multiple choice quizzes, video quizzes, detailed explanations

Additional Information

• To find the course, search "algebra" on Udemy, look for the black background image course.
• Course is suitable for enhancing understanding of multiple algebraic concepts.