Lesson 6: Negative Exponents

What does the negative exponent do to the expression/term?
what is the equivalent expression for x^-1?

HSN-RN.A.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Lesson Description This lesson focuses on the definition and manipulation of non-positive exponents. Students learn to navigate expressions where variables and numbers appear to have "negative power." Key skills include: The Negative Exponent Rule: Mastering the identity x^-n = 1/x^n and its counterpart 1/x^-n = x^n.The "Elevator" Analogy: Learning that moving a base between the numerator and denominator changes the sign of its exponent. Simplifying Complex Fractions: Applying the Product and Quotient rules in conjunction with negative exponents to ensure all final answers contain only positive exponents. Scientific Notation: Using negative exponents to represent extremely small decimals (e.g., 0.0005 = 5 \times 10^{-4}) and understanding their application in science and medicine. Zero as an Exponent: Reinforcing that x^0 = 1 (where x \neq 0) as the boundary between positive and negative growth.
Purpose
The purpose of Section 13.5 is to complete the student's understanding of the number line of exponents. It shifts the view of exponents from "repeated multiplication" to a "positional scale." This is critical for students entering health careers or lab sciences at Middlesex CC, as it provides the mathematical foundation for calculating dosages, understanding pH scales, and working with microscopic measurements in microbiology and chemistry.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): This section requires students to perform multi-step simplifications that are not always intuitive. A student must look at a term and decide on a sequence of operations (distributing the power, flipping the fraction, and simplifying coefficients). This involves coordinating multiple abstract rules to reach a standardized final form.

Do a variety of examples to help with the understanding of negative exponents.

"you try'' on page 312