56-60. Выполните действия: 56. \frac{(2+3i)-(5+7i)}{2+3i} 57. \frac{3+2i}{3-2i}+\frac{5+2i}{3+2i} 58. \frac{6+2i}{3-7i}-\frac{2+3i}{2+5i} 59. \frac{6+2i}{1-i} - i^{27} 60. i^{123}+(1-i)^6-(1+i)^8

Выполним указанные действия с комплексными числами.

1. \(\frac{(2+3i)-(5+7i)}{2+3i} = \frac{2+3i-5-7i}{2+3i} = \frac{-3-4i}{2+3i} = \frac{(-3-4i)(2-3i)}{(2+3i)(2-3i)} = \frac{-6+9i-8i+12i^2}{4-(3i)^2} = \frac{-6+i-12}{4+9} = \frac{-18+i}{13} = -\frac{18}{13} + \frac{1}{13}i\)
2. \(\frac{3+2i}{3-2i} + \frac{5+2i}{3+2i} = \frac{(3+2i)(3+2i)}{(3-2i)(3+2i)} + \frac{5+2i}{3+2i} = \frac{9+12i+4i^2}{9-(2i)^2} + \frac{5+2i}{3+2i} = \frac{9+12i-4}{9+4} + \frac{5+2i}{3+2i} = \frac{5+12i}{13} + \frac{5+2i}{3+2i} = \frac{5+12i}{13} + \frac{(5+2i)(3-2i)}{(3+2i)(3-2i)} = \frac{5+12i}{13} + \frac{15-10i+6i-4i^2}{9-(2i)^2} = \frac{5+12i}{13} + \frac{15-4i+4}{9+4} = \frac{5+12i}{13} + \frac{19-4i}{13} = \frac{5+12i+19-4i}{13} = \frac{24+8i}{13} = \frac{24}{13} + \frac{8}{13}i\)
3. \(\frac{6+2i}{3-7i} - \frac{2+3i}{2+5i} = \frac{(6+2i)(3+7i)}{(3-7i)(3+7i)} - \frac{(2+3i)(2-5i)}{(2+5i)(2-5i)} = \frac{18+42i+6i+14i^2}{9-(7i)^2} - \frac{4-10i+6i-15i^2}{4-(5i)^2} = \frac{18+48i-14}{9+49} - \frac{4-4i+15}{4+25} = \frac{4+48i}{58} - \frac{19-4i}{29} = \frac{2+24i}{29} - \frac{19-4i}{29} = \frac{2+24i-19+4i}{29} = \frac{-17+28i}{29} = -\frac{17}{29} + \frac{28}{29}i\)
4. \(\frac{6+2i}{1-i} - i^{27} = \frac{(6+2i)(1+i)}{(1-i)(1+i)} - i^{27} = \frac{6+6i+2i+2i^2}{1-i^2} - i^{27} = \frac{6+8i-2}{1+1} - i^{27} = \frac{4+8i}{2} - i^{27} = 2+4i - i^{27}\) \(i^{27} = i^{24+3} = i^{24} \cdot i^3 = (i^4)^6 \cdot i^3 = 1^6 \cdot i^3 = 1 \cdot i^3 = i^3 = -i\) \(2+4i - i^{27} = 2+4i - (-i) = 2+4i+i = 2+5i\)
5. \(i^{123}+(1-i)^6-(1+i)^8 = i^{120+3} + ((1-i)^2)^3 - ((1+i)^2)^4 = i^{120} \cdot i^3 + (1-2i+i^2)^3 - (1+2i+i^2)^4 = (i^4)^{30} \cdot i^3 + (1-2i-1)^3 - (1+2i-1)^4 = 1^{30} \cdot i^3 + (-2i)^3 - (2i)^4 = i^3 -8i^3 - 16i^4 = -i -8(-i) - 16(1) = -i+8i-16 = 7i-16 = -16+7i\)

Ответ:

1. \(-\frac{18}{13} + \frac{1}{13}i\)
2. \(\frac{24}{13} + \frac{8}{13}i\)
3. \(-\frac{17}{29} + \frac{28}{29}i\)
4. \(2+5i\)
5. \(-16+7i\)