Overview
This lecture explains how to evaluate exponents with negative bases, highlighting the importance of parentheses and how sign rules affect the result.
Exponents with Negative Bases: Parentheses Matter
- An exponent tells how many times to multiply the base by itself.
- Parentheses indicate the negative sign is part of the base (e.g., (-5)²).
- Without parentheses, the negative sign is not part of the base and stays in front (e.g., -5²).
Example 1: Negative Five Squared
- (-5)² = (-5) × (-5) = +25 because a negative times a negative is positive.
- -5² = -(5×5) = -25; negative sign stays in front due to order of operations.
Example 2: Negative Two Cubed
- (-2)³ = (-2) × (-2) × (-2) = +4 × -2 = -8; result is negative.
- -2³ = -1 × (2×2×2) = -1 × 8 = -8; same result as with parentheses.
Example 3: Negative Three to the Fourth Power
- (-3)⁴ = (-3) × (-3) × (-3) × (-3)
- (-3) × (-3) = +9; then × (-3) = -27; then × (-3) = +81.
Example 4: Negative Ten Cubed
- (-10)³ = (-10) × (-10) × (-10)
- (-10) × (-10) = +100; then × (-10) = -1000.
Key Terms & Definitions
- Exponent — A number indicating how many times the base is multiplied by itself.
- Base — The number being multiplied in an exponential expression.
- Parentheses — Symbols used to group terms and clarify which parts of an expression are included in operations.
- Order of Operations — Mathematical rules that determine the sequence in which operations are performed.
Action Items / Next Steps
- Practice evaluating exponents with both negative and positive bases, with and without parentheses.
- Remember to pay close attention to the use of parentheses in problems.