SECTION A.5 Rational Expressions A43 27. − + − − − x x x x x x 7 6 2 24 2 2 28. − − + + − x x x x x 3 1 5 24 2 29. − − + − x x x x 4 4 2 6 2 2 30. − − − − + x x x x x 3 1 4 2 1 2 31. ( ) ( ) ( )( ) − + + − + x x x x 3 1 1 2 1 1 2 2 32. ( ) ( ) ( )( ) + − − + − x x x x 2 2 1 6 2 1 2 2 33. + − x x 1 1 1 1 34. + − x x 4 1 3 1 2 2 35. − + + − + + − − x x x x x x x x 2 2 1 1 1 2 3 36. ( ) + − − − − + + x x x x x x x x 2 5 3 3 1 3 2 2 Applications and Extensions In Problems 37–44, expressions that occur in calculus are given. Reduce each expression to lowest terms. 37. ( ) ( ) ( ) + ⋅ − − ⋅ − x x x 2 3 3 3 5 2 3 5 2 38. ( ) ( ) ( ) + ⋅ − − ⋅ − x x x 4 1 5 5 2 4 5 2 2 39. ( ) ( ) ⋅ − + ⋅ + x x x x 2 1 1 1 2 2 2 40. ( ) ( ) ⋅ − − ⋅ − x x x x 2 4 1 4 2 2 2 41. ( ) ( ) + ⋅ − ⋅ + x x x x 3 1 2 3 3 1 2 2 42. ( ) ( ) − ⋅ − ⋅ − x x x x 2 5 3 2 2 5 2 3 2 43. ( ) ( ) ( ) + ⋅ − + ⋅ + x x x x 1 3 3 4 2 1 2 2 2 44. ( ) ( ) ( ) + ⋅ − − ⋅ + x x x x 9 2 2 5 2 9 2 2 2 45. The Lensmaker’s Equation The focal length f of a lens with index of refraction n is ( ) = − + f n R R 1 1 1 1 1 2 where R1 and R2 are the radii of curvature of the front and back surfaces of the lens. Express f as a rational expression. Evaluate the rational expression for = = n R 1.5, 0.1 meter, 1 and = R 0.2 meter. 2 46. Electrical Circuits An electrical circuit contains three resistors connected in parallel. If the resistance of each is R R , , 1 2 and R ohms, 3 respectively, their combined resistance R is given by the formula = + + R R R R 1 1 1 1 1 2 3 Express R as a rational expression. Evaluate R for = R 5 ohms, 1 = R 4 ohms, 2 and = R 10 ohms. 3 Explaining Concepts 47. The following expressions are called continued fractions: + + + + + + + + + + x x x x 1 1 , 1 1 1 1 , 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 , . . . Each simplifies to an expression of the form + + ax b bx c Trace the successive values of a, b, and c as you “continue” the fraction. Can you discover the patterns that these values follow? Go to the library and research Fibonacci numbers. Write a report on your findings. 48. Explain to a fellow student when you would use the LCM method to add two rational expressions. Give two examples of adding two rational expressions, one in which you use the LCM and the other in which you do not. 49. Which of the two methods given in the text for simplifying complex rational expressions do you prefer? Write a brief paragraph stating the reasons for your choice.
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