Overview
This lecture explains the concept of reciprocals and demonstrates how to simplify expressions with negative exponents using the reciprocal rule.
Reciprocals
- A reciprocal of a fraction swaps the numerator and denominator (e.g., reciprocal of 2/3 is 3/2).
- Multiplying a number by its reciprocal always results in 1.
- To find a reciprocal, reverse the positions of the numerator and denominator.
Negative Exponents
- Negative exponents indicate the reciprocal of the base raised to a positive exponent.
- It's best practice to rewrite expressions without negative exponents.
- The rule: ( x^{-n} = 1/x^n ).
Example Problems
- ( 7^{-2} = 1/7^2 = 1/49 ).
- ( (10/3)^{-3} ): Take the reciprocal to get ( (3/10)^3 = 27/1000 ).
- ( (-1.5)^{-4} ): Convert -1.5 to fraction ((-3/2)), take reciprocal ((2/-3)), raise to 4th power: (2^4/(-3)^4 = 16/81) (result is positive since exponent is even).
- ( (25/36)^{-1/2} ): Reciprocal is ( (36/25)^{1/2} ); square root of numerator and denominator is (6/5).
Key Terms & Definitions
- Reciprocal — The flipped version of a fraction; numerator and denominator are switched.
- Negative exponent — An exponent that signals the reciprocal of the base raised to the positive version of that exponent.
- Exponent — The power to which a number or expression is raised.
- Square root — A number that produces a specified quantity when multiplied by itself.
Action Items / Next Steps
- Practice rewriting expressions with negative exponents as reciprocals.
- Review previous lesson on fractional exponents (e.g., ( x^{1/2} ) as square root).
- Complete assigned exercises on negative exponents and reciprocals.