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Algebra 2: Section 7.1 Nth Roots and Rational Exponents 1 7.1: Evaluating nth Roots • You are familiar with square roots – 2 is the square root of 4, since 22 = 4. • This concept can be extended to other types of roots – 2 is the cube root of 8, since 23 = 8 – 2 is the fourth root of 16, since 24 = 16 – 2 is the fifth root of 32, since 25 = 32 – 2 is the nth root of a, since 2n = a 1 n Rational Exponents a a n m m n a ( a) a m 1 1 n m n m a ( a ) n a m n 1 n 3 Examples • Rewrite using rational exponents. 1. 2. 3 27 27 5 a a 1 3 3. 7 4. 3 5 1 5 8 2 7 5 2 2 8 3 3 • Rewrite using radical notation. 5. 7 4 3 7 3 4 4 5 6. 4 4 5 4 4 1 4 Nth Roots Rules for Signs of Answers • For odd roots (3, 5, 7, etc) – There is ALWAYS ONLY ONE answer (either + or -) • For even roots (2, 4, 6, etc) – If the radical symbol is already in the problem, then use the sign that is in front – If YOU put the radical symbol in the problem to solve an equation, then there will be two answers (+ and -) 5 Examples • Find the nth root(s) of a 7. n = 5, a = -32 -2 What multiplied by itself 5 times equals -32? 8. n = 3: a = 64 4 What multiplied by itself 3 times equals 64? 9. n = 4: a = 256 4 What multiplied by itself 4 times equals 256? 10. n = 2: a = 169 ±13 What multiplied by itself 2 times equals 169? 6 Examples Simplify or evaluate each expression: 3 11. 5 125 12. 13. 0 4 6 16 2 3 2 2 0 3 4 0 7 Rational Exponents (examples) • Simplify or evaluate each expression: 14. 15. 16. 82/3 163/4 4-5/2 3 2 1 4 3 16 4 2 5 4 64 3 8 2 8 1 1 32 3 2 5 Examples • Solve the equation. Round your answer to two decimal places. 3 17. 5x = 40 x 8 3 x2 9 Examples 18. ( y 1) 32 3 y 1 32 3 3 19. 3(x +5) = 81 x 5 3 27 y 32 1 x 5 3 4.17 x 2 3 10 Homework P.404 #13-22 all #24-40 evens #53-61 all 11 List of Perfects Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 Perfect Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 Perfect Quads: 1, 16, 81, 256, 625 Perfect Fifths: 1, 32, 243, 1024, 3125