Lesson Description
When working with exponents, negative bases can change the sign of the result depending on whether the exponent is even or odd. This lesson explains how to evaluate expressions like (-3)^2 versus -3^2 and why parentheses make a big difference.
Learning Objectives
- Evaluate powers with negative bases.
- Determine whether results are positive or negative using even and odd exponents.
- Understand how parentheses affect exponent calculations.
Topics Covered
- Even vs. odd exponents
- Difference between (-a)^n and -a^n
- Order of operations with exponents
Key Terms
- Negative Base: A number less than zero being raised to a power.
- Even Exponent: An exponent divisible by 2, producing a positive result when applied to a negative base.
- Odd Exponent: An exponent not divisible by 2, keeping the negative sign when applied to a negative base.
Lesson Examples
Example 1: Evaluate (-4)^2
- The base is -4 because of the parentheses.
- Square -4: (-4) × (-4).
- The result is 16 (negative times negative equals positive).
Example 2: Evaluate -4^2
- No parentheses, so only 4 is squared.
- 4^2 = 16.
- Apply the negative sign: -16.
Practice Problems
- Evaluate (-5)^2.
- Evaluate (-5)^3.
- Evaluate -6^2.
- Evaluate (-2)^4.
Tips & Common Mistakes
- Always check for parentheses.
- Even exponents make negative bases positive.
- Without parentheses, the exponent applies only to the number, not the negative sign.
Summary
Exponents with negative bases behave differently depending on whether the exponent is even or odd. Parentheses determine whether the negative sign is part of the base. Carefully apply order of operations to avoid sign errors.
Challenge Problems
- Evaluate (-3)^5.
- Compare (-7)^2 and -7^2. Explain the difference.
- Simplify (-2)^3 × (-2)^2.