If z = 3 + i 2 then z 101 + i 103 105 =

Step-by-step solution

Given, z = 3 + i 2 = 3 2 + i 1 2 = cos π 6 + i sin π 6 Then, z 101 = cos π 6 + i sin π 6 101 = cos 101 π 6 + i sin 101 π 6 = cos 101 6 π + i sin 101 6 π = cos 17 π - π 6 + i sin 17 π - π 6 = - cos π 6 + i sin π 6 = - 3 + i 2 And, i 103 = i 4 × 25 + 3 = i 3 = - i Therefore, z 101 + i 103 105 = - 3 2 + i 2 - i 105 = - 3 2 - i 2 105 = - 3 + i 2 105 = - z 105 Now, - z 105 = - 3 + i 2 105 = - cos π 6 + i sin π 6 105 = - cos 105 π 6