Let α , β be the roots of the equation x 2 - 2 x + 2 = 0 Then α 14 + β 14 is equal to

Step-by-step solution

Given, x 2 - 2 x + 2 = 0 ⇒ x = 2 ± 6 i 2 ⇒ x = 2 1 ± 3 i 2 ⇒ x = 2 1 2 ± 3 i 2 ⇒ x = 2 cos π 3 ± i sin π 3 So, α = 2 cos π 3 + i sin π 3 = 2 e iπ 3 And β = 2 cos π 3 - i sin π 3 = 2 e - iπ 3 Now α 14 + β 14 = 2 e iπ 3 14 + 2 e - iπ 3 14 ⇒ α 14 + β 14 = 2 14 e i 14 π 3 + e - i 14 π 3 ⇒ α 14 + β 14 = 2 14 cos 14 π 3 + i sin 14 π 3 + cos 14 π 3 - i s