44-55. Выполните деление: 44. \frac{5i}{3+2i} 45. \frac{-2i}{5-i} 46. \frac{2-3i}{5+2i} 47. \frac{3-i}{5-3i} 48. \frac{3+2i}{1-5i} 49. \frac{3-7i}{3+2i} 50. \frac{3+2i}{5i} 51. \frac{6-7i}{i} 52. \frac{2+3i}{2-3i} 53. \frac{5-7i}{5+7i} 54. \frac{1-i}{1+i} 55. \frac{1+i}{1-i}

Выполним деление комплексных чисел, умножив числитель и знаменатель каждой дроби на комплексно сопряженное знаменателю число.

1. \(\frac{5i}{3+2i} = \frac{5i(3-2i)}{(3+2i)(3-2i)} = \frac{15i - 10i^2}{9 - (2i)^2} = \frac{15i + 10}{9 + 4} = \frac{10+15i}{13} = \frac{10}{13} + \frac{15}{13}i\)
2. \(\frac{-2i}{5-i} = \frac{-2i(5+i)}{(5-i)(5+i)} = \frac{-10i - 2i^2}{25 - i^2} = \frac{-10i + 2}{25 + 1} = \frac{2-10i}{26} = \frac{1}{13} - \frac{5}{13}i\)
3. \(\frac{2-3i}{5+2i} = \frac{(2-3i)(5-2i)}{(5+2i)(5-2i)} = \frac{10 - 4i - 15i + 6i^2}{25 - (2i)^2} = \frac{10 - 19i - 6}{25 + 4} = \frac{4 - 19i}{29} = \frac{4}{29} - \frac{19}{29}i\)
4. \(\frac{3-i}{5-3i} = \frac{(3-i)(5+3i)}{(5-3i)(5+3i)} = \frac{15 + 9i - 5i - 3i^2}{25 - (3i)^2} = \frac{15 + 4i + 3}{25 + 9} = \frac{18 + 4i}{34} = \frac{9}{17} + \frac{2}{17}i\)
5. \(\frac{3+2i}{1-5i} = \frac{(3+2i)(1+5i)}{(1-5i)(1+5i)} = \frac{3 + 15i + 2i + 10i^2}{1 - (5i)^2} = \frac{3 + 17i - 10}{1 + 25} = \frac{-7 + 17i}{26} = -\frac{7}{26} + \frac{17}{26}i\)
6. \(\frac{3-7i}{3+2i} = \frac{(3-7i)(3-2i)}{(3+2i)(3-2i)} = \frac{9 - 6i - 21i + 14i^2}{9 - (2i)^2} = \frac{9 - 27i - 14}{9 + 4} = \frac{-5 - 27i}{13} = -\frac{5}{13} - \frac{27}{13}i\)
7. \(\frac{3+2i}{5i} = \frac{(3+2i)(-5i)}{5i(-5i)} = \frac{-15i - 10i^2}{-25i^2} = \frac{-15i + 10}{25} = \frac{10-15i}{25} = \frac{2}{5} - \frac{3}{5}i\)
8. \(\frac{6-7i}{i} = \frac{(6-7i)(-i)}{i(-i)} = \frac{-6i + 7i^2}{-i^2} = \frac{-6i - 7}{1} = -7 - 6i\)
9. \(\frac{2+3i}{2-3i} = \frac{(2+3i)(2+3i)}{(2-3i)(2+3i)} = \frac{4 + 12i + 9i^2}{4 - (3i)^2} = \frac{4 + 12i - 9}{4 + 9} = \frac{-5 + 12i}{13} = -\frac{5}{13} + \frac{12}{13}i\)
10. \(\frac{5-7i}{5+7i} = \frac{(5-7i)(5-7i)}{(5+7i)(5-7i)} = \frac{25 - 70i + 49i^2}{25 - (7i)^2} = \frac{25 - 70i - 49}{25 + 49} = \frac{-24 - 70i}{74} = -\frac{12}{37} - \frac{35}{37}i\)
11. \(\frac{1-i}{1+i} = \frac{(1-i)(1-i)}{(1+i)(1-i)} = \frac{1 - 2i + i^2}{1 - i^2} = \frac{1 - 2i - 1}{1 + 1} = \frac{-2i}{2} = -i\)
12. \(\frac{1+i}{1-i} = \frac{(1+i)(1+i)}{(1-i)(1+i)} = \frac{1 + 2i + i^2}{1 - i^2} = \frac{1 + 2i - 1}{1 + 1} = \frac{2i}{2} = i\)

Ответ:

1. \(\frac{10}{13} + \frac{15}{13}i\)
2. \(\frac{1}{13} - \frac{5}{13}i\)
3. \(\frac{4}{29} - \frac{19}{29}i\)
4. \(\frac{9}{17} + \frac{2}{17}i\)
5. \(-\frac{7}{26} + \frac{17}{26}i\)
6. \(-\frac{5}{13} - \frac{27}{13}i\)
7. \(\frac{2}{5} - \frac{3}{5}i\)
8. \(-7 - 6i\)
9. \(-\frac{5}{13} + \frac{12}{13}i\)
10. \(-\frac{12}{37} - \frac{35}{37}i\)
11. \(-i\)
12. \(i\)