Fractions and Negative Numbers: Parentheses are crucial when the base is a fraction or a negative number.
Examples:
(9/7)^4 ensures the entire fraction is used as the base.
(-2)^6 ensures -2 is the base, not just 2 with a negative sign outside.
Special Powers
X^1: X
X^n: X * X * ... * X (n times)
X^2: X squared (Example: area of a square)
X^3: X cubed (Example: volume of a cube)
Exercises
Convert repeated multiplication into exponential notation.
Identify and use parentheses correctly in expressions.
Recognize when to use negative and positive results with exponents.
Negative Base and Patterns
Key Point: Whether the product is positive or negative depends on if the exponent is even or odd.
Examples:
(-1)^12 - Positive (even exponent)
(-1)^13 - Negative (odd exponent)
(-5)^95 - Negative (95 is odd)
(-1.8)^122 - Positive (122 is even)
Patterns with Negative Numbers
Even Exponent: Result is positive.
Odd Exponent: Result is negative.
Understanding Expressions
Josie Example: -15^6 vs. (-15)^6 - Parentheses make a significant difference.
Use parentheses for clarity: Properly indicate the base, especially with negative numbers and fractions.
Real-World Application: Half-Life
Concept: Substance decays by half every set period.
Example: After n periods (half-lives), remaining substance can be written using exponential notation.
Example: 500 grams decaying with a half-life.
1 period: (1/2) * 500 grams
2 periods: (1/2)^2 * 500 grams
n periods: (1/2)^n * starting mass
Summary
Goals: Recognize the structure and meaning of exponential expressions, understand the significance of parentheses, apply these concepts to practical problems like half-life.
Exercises: Practice with various bases and exponents, interpret results, and solidify understanding. Make corrections regularly for consistent learning.
Next Steps
Exit Ticket: Test understanding of the concepts learned.
Problem Set: Apply learned concepts to further examples and exercises.
Regular Corrections: Check and correct work daily to ensure understanding.