3 + i = 2 ( cos π 6 + i sin π 6 ) \displaystyle \sqrt{3} + i = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) 3 + i = 2 ( cos 6 π + i sin 6 π ) ( 3 + i ) 108 = 2 108 ( cos 108 ⋅ π 6 + i sin 108 ⋅ π 6 ) \displaystyle ( \sqrt{3} + i ) ^{108} = 2 ^{108} \left( \cos 108 \cdot \frac{\pi}{6} + i \sin 108 \cdot \frac{\pi}{6} \right) ( 3 + i ) 108 = 2 108 ( cos 108 ⋅ 6 π + i sin 108 ⋅ 6 π ) = 2 108 ( cos 18 π + i sin 18 π ) = 2 108 = 2 ^{108} \left( \cos 18 \pi + i \sin 18 \pi \right) = 2 ^{108} = 2 108 ( cos 18 π + i sin 18 π ) = 2 108 3 − i = 2 { cos ( − π 6 ) + i sin ( − π 6 ) } \displaystyle \sqrt{3} - i = 2 \left\{ \cos \left( - \frac{\pi}{6} \right) + i \sin \left( - \frac{\pi}{6} \right) \right\} 3 − i = 2 { cos ( − 6 π ) + i sin ( − 6 π ) } ( 3 − i ) 108 = 2 108 { cos ( − 18 π ) + i sin ( − 18 π ) } = 2 108 \displaystyle ( \sqrt{3} - i ) ^{108} = 2 ^{108} \left\{ \cos \left( - 18 \pi ) + i \sin \left( - 18 \pi ) \right\} = 2 ^{108} \right. \right. ( 3 − i ) 108 = 2 108 { cos ( − 18 π ) + i sin ( − 18 π ) } = 2 108 ∴ ( 3 + i ) 108 + ( 3 − i ) 108 = 2 108 + 2 108 \displaystyle \therefore ( \sqrt{3} + i ) ^{108} + ( \sqrt{3} - i ) ^{108} = 2 ^{108} + 2 ^{108} ∴ ( 3 + i ) 108 + ( 3 − i ) 108 = 2 108 + 2 108 = 2 ⋅ 2 108 = 2 109 = 2 \cdot 2 ^{108} = 2 ^{109} = 2 ⋅ 2 108 = 2 109