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Lesson 36 Section 7.2 1 2 What is a value for 9 ? Consider This. 1 2 1 2 1 1 2 2 9 9 9 91 9 using the product rule of exponents. Now, Think! What other number times itself equals 9? 3 3 9 1 2 Since both products equal 9, we can conclude that 9 3 . Conclusion: A rational exponent is the same as a root! 1 Definition of a rational (fractional) exponent: a n n a Evaluate or write an equivalent root for each. 1 2 1. 16 1 3 2. x 1 5 3. (32) 1 4. (4c) 4 Write an equivalent expression using exponential notation. 5. 4 5 6. 3 ab 2 7. n Expanded Definition of a rational (fractional) exponent: a m n n a m or a n m Notice: The exponent n can be inside the radical (in the radicand) or outside of the radical. The order does not matter. You can raise to the m power first and then take the nth root. Or you can take the nth root and then raise to the m power. 1 The index of the root comes from the denominator. The exponent is the numerator. Evaluate or write an equivalent root for each. 2 8. 27 3 9. * 16 3 2 3 10. (mn) 4 Write an equivalent expression using exponential notation. 11. 12. 83 4 (ab 3 5 The *rules of exponents we had back at the beginning of the semester also apply with these rational exponents. Here is a summary of those rules. Product Rule: a m a n a m n am Quotient Rule: n a m n a Negative Exponent Rule: a a mn ab n a nb n Power Rules: a m n n 1 1 a n or n a n or a a b n b a n n an a bn b Use the rules of exponents to simplify. 2 5 13. 8 3 8 3 14. 2 2 5 8 1 8 2 1 4 15. x 4 x 5 2 16. 4 3 9 2 1 2 3 2 17. 4 3 m 4 13 18. x 4 3