Lecture on Exponent Rules

Overview

• Recap of exponent rules.
• Application of the four main exponent rules.
• Practice problems combining all rules.

Exponent Rules

Rule 1: Multiplication of Same Bases

• Example: (2^3 \times 2^5)
  • Process: Add exponents when multiplying like bases.
  • Result: (2^{3+5} = 2^8)

Rule 2: Division of Same Bases

• Example: (3^8 \div 3^2)
  • Process: Subtract exponents when dividing like bases.
  • Result: (3^{8-2} = 3^6)

Rule 3: Power to a Power

• Example: ((3^2)^3)
  • Process: Multiply the exponents.
  • Result: (3^{2 \times 3} = 3^6)

Rule 4: Zero Exponent

• Concept: Any base raised to the power of 0 equals 1.
• Example: (a^0 = 1)

Practice Problems

Example 1

• Expression: (2^8 \div 2^3 \times 2^4)
  • Step 1: Apply Rule 2: (2^{8-3} = 2^5)
  • Step 2: Apply Rule 1: (2^5 \times 2^4 = 2^{5+4} = 2^9)

Example 2

• Expression: ((3^4 \times 3^2) \div 3^5)
  • Step 1: Apply Rule 1: (3^{4+2} = 3^6)
  • Step 2: Apply Rule 2: (3^6 \div 3^5 = 3^{6-5} = 3^1)

Example 3

• Expression: ((3^2)^4 \times 3^2)
  • Step 1: Inside the bracket: Apply Rule 3: (3^{2 \times 4} = 3^8)
  • Step 2: Outside the bracket: Apply Rule 1: (3^8 \times 3^2 = 3^{8+2} = 3^{10})

Example 4

• Expression: (3^5 \div 3^2)
  • Step: Apply Rule 2: (3^5 \div 3^2 = 3^{5-2} = 3^3)

Example 5

• Expression: (2^6 \times 2^3 \times 2^4 \div (2^5)^2)
  • Step 1: Apply Rule 1: (2^{6+3+4} = 2^{13})
  • Step 2: Apply Rule 3: ((2^5)^2 = 2^{5 \times 2} = 2^{10})
  • Step 3: Apply Rule 2: (2^{13} \div 2^{10} = 2^{13-10} = 2^3)

Conclusion

• Emphasis on understanding and practicing different methods.
• Encouragement to apply the rules in different combinations for simplification.
• Importance of mastering these rules for problem-solving in algebra.