Learning Goal
Part of: Extend the properties of exponents to rational exponents — 1 of 2 cluster items
Explain rational exponents
**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.
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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.
What you'll learn
- 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
Slides
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Slide Video
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