Lesson Plan

Explain why 5^(1/3) must equal the cube root of 5, using the reasoning that (5^(1/3))³ must equal 5^((1/3)·3) = 5¹ = 5

Extend integer exponent properties (product rule, power rule, quotient rule) to rational exponents and justify each extension

Interpret rational exponents as radicals: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)

Evaluate expressions with rational exponents by converting between exponential and radical notation

Explain in their own words why the definition of rational exponents is not arbitrary but is forced by the requirement that exponent properties remain consistent

HSN.RN.A.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. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))³ = 5^((1/3)·3) to hold, so (5^(1/3))³ must equal 5.