Understanding and Applying Exponent Rules

May 11, 2025

Exponent Rules Summary and Application

Overview

  • Review of exponent rules.
  • Application of all four rules together in examples.
  • Importance of understanding each rule for effective problem-solving.

Exponent Rules

Rule 1: Multiplying with Same Base

  • Example: (2^3 \times 2^5)
  • Rule: When bases are the same, add exponents.
  • Result: (2^{3+5} = 2^8)

Rule 2: Dividing with Same Base

  • Example: (3^8 \div 3^2)
  • Rule: When bases are the same, subtract exponents.
  • Result: (3^{8-2} = 3^6)

Rule 3: Power of a Power

  • Example: ((3^2)^3)
  • Rule: Multiply the exponents.
  • Result: (3^{2\times3} = 3^6)

Rule 4: Zero Exponent

  • Rule: Anything raised to the power of zero equals 1.
  • Example: (a^0 = 1)

Applying Exponent Rules

Example 1

  • Expression: (2^8 \div 2^3)
  • Process:
    • Division (Rule 2) (\rightarrow 2^{8-3} = 2^5)

Example 2

  • Expression: (2^8 \times 2)
  • Process:
    • Multiplication (Rule 1) (\rightarrow 2^{8+1} = 2^9)

Example 3

  • Complex Expression: ((3^4 \times 3^2) \div 3^5)
  • Process:
    • Inside Bracket: Rule 1 (\rightarrow 3^{4+2} = 3^6)
    • Apply Rule 2: (3^6 \div 3^5 = 3^{6-5} = 3^1)
    • Final: Apply Rule 3 on remaining expression

Example 4

  • Expression: (3^{4\times3} \times (3^2 \div 3^5))
  • Process:
    • Simplify using Rule 2 and Rule 1
    • Result: (3^7)

Example 5

  • Expression: (3^0 \times 4)
  • Process:
    • Apply Rule 4: (3^0 = 1)
    • Simplifies to: (1 \times 4 = 4)

Example 6

  • Expression: ((2^6 \times 2^3 \times 2^4) \div (2^5)^2)
  • Process:
    • Use Rule 1 for numerator (2^{6+3+4} = 2^{13})
    • Apply Rule 3 to denominator ((2^5)^2 = 2^{10})
    • Final: Use Rule 2 (2^{13-10} = 2^3)

Conclusion

  • Always keep the base the same across operations.
  • Use rules systematically to simplify expressions.
  • Practice with examples to strengthen understanding and application of exponent rules.