837 8.6 De Moivre’s Theorem; Powers and Roots of Complex Numbers Solve each problem. 49. Solve the cubic equation x3 = 1 by writing it as x3 - 1 = 0, factoring the left side as the difference of two cubes, and using the zero-factor property. Apply the quadratic formula as needed. Then compare the solutions to those of Exercise 37. 50. Solve the cubic equation x3 = -27 by writing it as x3 + 27 = 0, factoring the left side as the sum of two cubes, and using the zero-factor property. Apply the quadratic formula as needed. Then compare the solutions to those of Exercise 42. 51. Mandelbrot Set The fractal known as the Mandelbrot set is shown in the figure. To determine whether a complex number z = a + bi is in this set, perform the following sequence of calculations. Repeatedly compute z, z2 + z, 1z2 + z22 + z, 31z2 + z22 + z42 + z, . . . . In a manner analogous to the Julia set, the complex number z does not belong to the Mandelbrot set if any of the resulting absolute values exceeds 2. Otherwise z is in the set, and the point 1a, b2 should be shaded in the graph. Determine whether the following numbers belong to the Mandelbrot set. (Data from Lauwerier, H., Fractals, Princeton University Press.) (a) z = 0 + 0i (b) z = 1 - 1i (c) z = -0.5i For each of the following, (a) find all cube roots of each complex number. Write answers in trigonometric form. (b) Graph each cube root as a vector in the complex plane. See Examples 2 and 3. 19. cos 0° + i sin0° 20. cos 90° + i sin90° 21. 8 cis 60° 22. 27 cis 300° 23. -8i 24. 27i 25. -64 26. 27 27. 1 + i23 28. 2 - 2i23 29. -223 + 2i 30. 23 - i Find and graph all specified roots of 1. 31. second (square) 32. fourth 33. sixth Find and graph all specified roots of i. 34. second (square) 35. third (cube) 36. fourth Find all complex number solutions of each equation. Write answers in trigonometric form. See Example 4. 37. x3 - 1 = 0 38. x3 + 1 = 0 39. x3 + i = 0 40. x4 + i = 0 41. x3 - 8 = 0 42. x3 + 27 = 0 43. x4 + 1 = 0 44. x4 + 16 = 0 45. x4 - i = 0 46. x5 - i = 0 47. x3 - A4 + 4i23B = 0 48. x4 - A8 + 8i23B = 0
RkJQdWJsaXNoZXIy NjM5ODQ=