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Rational Numbers as Exponents Section 7-5 Objectives • To calculate radical expressions in two ways. • To write expressions with rational exponents as radical expressions and vice versa. • To simplify expressions containing negative rational exponents. • To use rational exponents to simplify radical expressions. Simplify and Compare 3 2 8 and 4 8 and 4 2 3 2 2 Theorem 7-6 For any nonnegative number a, any natural number index k , and any integer m, k a m a k m . We can raise to a power and then take a root or we can take a root and then raise to a power. 3 27 729 9 2 3 27 3 2 3 9 2 2 64 4 3 6 2 3 3 6 2 2 2 2 2 2 8 8 4 3 3 5x 5x 3 3 3 3 3 3 3 3 5x 5x 5x 5x 5x (5x) 2 5x 5x 5x 3 8 2 6y 3 Definition For any nonnegative number a and any 1 ak natural number index k , means k a (the nonnegative kth root of a). 1 2 x x 1 3 27 27 3 abc 7 3 1 5 abc 5 7 xy 7 xy 1 7 Definition For any natural numbers m and k , and any nonnegative number a, a k m . m ak k m means a or 27 4 3 2 2 3 3 27 4 2 3 or 27 3 or 4 3 2 Write with rational exponents 3 4 8 8 4 7xy 5 4 3 7xy 5 4 What is Write as a radical 9 1 2 7 1 2 x 3 Write with Rational Exponents 1 3 11 8 5 3x 1 3 Numerator Exponent Deno min ator Index / Root 1 3 1 2 x x x x 2 6 1 3 3 x ( x) x ( x) 2 3 2 6 3 2 3 3 1 x ( x ) x ( x ) x ( x) 3 2 3 3 http://www.youtube.com/watch?v=-T1punCdxas Negative Rational Exponents m For any rational number and any k positive real number a, 1 m. ak m a k means 1 2 4 1 4 5xy 4 5 1 2 1 5xy 4 5 Variables w/ Exponents • Reminder: When Multiplying Expressions with similar Bases the Exponents ADD! • When Dividing Expressions with similar Bases the Exponents SUBTRACT! x *x x 2 3 3 23 x 5 x 3 2 1 x x x 2 x Variables w/ Exponents • When Taking an Exponential of an Expression with a similar Base the Exponents MULTIPLY! • Reminder: Negative Exponents mean you TAKE THE RECIPROCAL! (x ) x 2 x 2 3 2*3 x 6 1 1 2 or 2 x 2 x x 2 5 3 5 3 3 3 16 16 1 4 1 2 16 1 1 4 2 3 4 2 3 5 5 16 3 5 5 1 4 72 72 72 2 3 6 12 1 2 Using Rational Exponents to Simplify Radical Expressions • Convert radical expressions to exponential expressions. • Use the properties of exponents to simplify. • Convert back to radical notation when appropriate. Use Rational Exponents to Simplify 6 6 x 3 4 3 6 x 1 6 4 1 2 x 1 2 6 2 x 2 6 2 1 3 2 2 3 Use Rational Exponents to Simplify 4 6 2 4 1 2 a a a a 2 12 6 6 6 2 a 2 a 2 a 4a 12 6 1 2 2 1 4 2 4 10 3 10 3 10 3 4 1 4 10 3 300 4 2 4 Use Rational Exponents to Simplify 4 x y x y 3 x y 3 4 x y 1 2 x y 3 1 4 2 1 4 x y 4 x y Write as a Single Radical Expression x 2 4 3y 1 2 x 2 3 y 1 4 2 4 1 4 x 2 3 y 1 4 x 2 3 y 2 4 4 x 2 3 y 2 4 x 3x y 12 xy 12 y 2 2 4x 4 3 y Write as a Single Radical Expression 1 2 1 2 5 6 a b c 3 6 3 6 5 6 3 3 5 a b c ab c 6 Write as a Single Radical Expression 2 3 1 2 5 6 4 6 3 6 5 6 x y z 4 x y z x y z 6 3 5 Methods to Simplify Radical Expressions • Simplify by factoring • Rationalize denominators • Collecting like radical terms • Using rational exponents Now you can • Calculate radical expressions in two ways. • Write expressions with rational exponents as radical expressions and vice versa. • Simplify expressions containing negative rational exponents. • Use rational exponents to simplify radical expressions. 1. 6a 3 6a 3 6a 6a 6a 6a 6a 7. 3 2 3 4 3 d 3 3 3 12c d 3 144 c 3 3 2 3 8 18 c 3 2c 18cd 2 12c d 2 2 2 c 3 d 2 13. 19. 25. 1 2 2 5 ab 1 5 3 2 at 3 5 19 19 2 2 ab 5 3 at 1 3 31. 37. 7 3 7 2 7 2 7 3 6 3 6 x y z x y z 3 6 2 2 3 12ab 1 2 1 2 1 2 3 6 12 a b 12 a b 12ab 1 2 40. x 1 3 1 x 43. 46. 5xy 1 x 2 3 5 6 1 3 1 x 5xy 2 3 5 6 2 3 1 2 2 1 3 2 49. 11 11 11 52. 8.3 8.3 3 4 2 5 8.3 3 2 4 5 4 3 6 6 11 15 8 20 20 8.3 7 6 11 8.3 7 20 55. 58. 3 5 7 54 3 8y 6 5 5 3 47 1 3 6 3 5 15 28 8 y 2y 2 61. 4 16x12 y16 4 4 12 4 16 x 12 16 2x 4 y 4 16 y 2x y 3 4 1 3 1 2 x x 2 x ( x 2) 3 64. 3 6 2 6 x ( x 2) x ( x 2) 6 6 3 2 x ( x 4 x 4) ( x 4 x 4 x ) 3 2 6 5 4 3 3 ( x y) 2 4 ( x y) 3 67. ( x y) ( x y) 2 3 3 4 ( x y) 12 ( x y) 8 9 12 12 ( x y) 1 2 3 3 4 ( x y) 1 12 7 12 5 6 1 3 1 6 s t 70. s t 3 12 s t 4 st 4 12 12 s 7 4 12 12 1 4 4 4 t s t 5 1 6 6