If the complex cube roots of   ( - i ) are α ,   β ,   γ , then α 2 + β 2 + γ 2 =

Step-by-step solution

- i 1 3 = α ,   β ,   γ α 2 + β 2 + γ 2 = ? - i = 1 and argument is = - π 2 ⇒ So we can change it in polar form cos 2 m π - π 2 + i sin 2 m π - π 2 1 3 ⇒ cos 2 m π 3 - π 6 + i sin 2 m π 3 - π 6 putting m = 0 ,   1 ,   2 ⇒ cos - π 6 + i sin - π 6 ⇒ 3 2 - i 1 2 = α Now putting m = 1 cos 2 π 3 - π 6 + i sin 2 π 3 - π 6 ⇒ cos π 2 + i sin π 2 ⇒ i