Exponent Rules Lecture Notes

Learning Goals

• Develop an understanding of exponent rules.
• Learn how to apply these rules to simplify expressions with exponents.

Key Terminology

• Base: The number that is being multiplied (e.g., 2 in (2^5)).
• Exponent: Indicates how many times the base is multiplied by itself (e.g., 5 in (2^5)).
• Power: The entire expression involving the base and exponent (e.g., (2^5)).

Exponent Rules

Multiplying Powers with the Same Base

• Rule: Add the exponents.
  • Example: (5^5 \times 5^3 = 5^{5+3} = 5^8)

Dividing Powers with the Same Base

• Rule: Subtract the exponents.
  • Example: (5^5 / 5^3 = 5^{5-3} = 5^2)

Power Raised to Another Exponent

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

Application of Exponent Rules

• Example 1: Simplifying (x^4 \times x^6)

  • Add exponents: (x^{4+6} = x^{10})

• Example 2: Simplifying (t^8 / t)

  • Subtract exponents (invisible 1 in the divisor): (t^{8-1} = t^7)

• Example 3: ((b^3)^5 \times b^2)

  • Power to a power first, then multiply:
    • ((b^3)^5 = b^{3 \times 5} = b^{15})
    • (b^{15} \times b^2 = b^{15+2} = b^{17})

• Example 4: (a^5^2 / a)

  • Power to a power first, then divide:
    • (a^{5 \times 2} = a^{10})
    • (a^{10} / a^1 = a^{10-1} = a^9)

Summary of Exponent Rules

1. Multiplication: Add exponents.
2. Division: Subtract exponents.
3. Power of a power: Multiply exponents.

Practice Problems

• Simplify using the appropriate rules:
  • (a^4 \times a^3 = a^7)
  • ((x^3 \times x^4) / x = x^{7-1} = x^6)
  • ((y^3)^2 \times y^4 = y^{6+4} = y^{10})
  • ((d^4)^3 / (d^2 \times d^9) = d^{12-11} = d^1 = d)

Conclusion

• The most challenging part of working with exponents is determining the correct operation: adding, subtracting, or multiplying exponents based on the operation being performed.
• Master the rules to simplify expressions efficiently and accurately.