Question

计算（1+根号3i/1-根号3i）^9

Answer 1

解析：
(1-根号3i)^10
=2^10 ×(1/2 -根号3/2 ×i)^10
=2^10 ×[cos(-π/3)+sin(-π/3)×i]^10
=2^10 ×[cos(-10π/3)+sin(-10π/3)×i]
=2^10 ×[cos(2π/3)+sin(2π/3)×i]
=2^10 ×（-1/2+根号3/2×i ）
而 (1+根号3i)^10
=2^10 ×(1/2 +根号3/2 ×i)^10
=2^10 ×[cos(π/3)+sin(π/3)×i]^10
=2^10 ×[cos(10π/3)+sin(10π/3)×i]
=2^10 ×[cos(4π/3)+sin(4π/3)×i]
=2^10 ×[-cos(π/3)-sin(π/3)×i]
=2^10 ×（-1/2-根号3/2×i ）
所以：
(1-根号3i)^10-(1+根号3i)^10
=2^10 ×（-1/2+根号3/2×i ）－2^10 ×（-1/2-根号3/2×i ）
=2^10 ×[（-1/2+根号3/2×i ）-（-1/2-根号3/2×i ）]
=2^10 ×根号3×i
=1024根号3×i