Lesson 3: Properties of Exponents

1. Look at the exponent laws. Which ones are similar?
2. Redraw the Concept Summary as a spider chart or concept chart in any shape that makes sense to you. Write each exponent law and at least one example of each type.
3. Expand your concept summary by including an example of each exponent law using small real numbers for the base and exponent. How does using real numbers help you to make sense of the exponent laws?

A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Lesson Description This lesson focuses on mastering the rules that govern exponential notation. Rather than calculating large numerical values, students learn to manipulate the symbols themselves. Key properties include: The Product Rule: Understanding that when multiplying like bases, you add the exponents: x^a times x^b = x^{a+b}.The Quotient Rule: Understanding that when dividing like bases, you subtract the exponents
The Power Rule: Learning that raising a power to a power requires multiplication
Power of a Product/Quotient: Distributing exponents to every factor inside parentheses. Zero Exponent Rule: Discovering why any non-zero base raised to the power of zero equals 1 (x^0 = 1). Negative Exponent Rule: Understanding that a negative exponent represents a reciprocal
Purpose
The purpose of Section 13.3 is to promote mathematical efficiency. Exponents are a shorthand for repeated multiplication; the properties of exponents are the shorthand for simplifying that multiplication. Without these rules, scientific notation and the study of exponential growth (common in Finance and Biology) would be impossible to manage. This section provides the "mental tools" needed to handle the complex polynomial multiplication and division that follow in the curriculum.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): This section moves beyond simple recall. Students must examine a complex expression and determine which rule to apply and in what order. For example, a problem might require the Power Rule followed by the Quotient Rule. The student must demonstrate a flexible understanding of these rules to simplify an expression into its most "elegant" form, often requiring the management of both coefficients and multiple variables simultaneously.

Have students write their own summary of the properties of powers, such as "to multiply expressions with exponents, you add the exponents; to divide, you subtract the exponents."

"you try" on page 306 and practice problems on page 317-318 #2 all