已知复数z=r(cosθ+isinθ)r,θ∈R (1)分别计算z^2,z^3并由此可归纳出z^n

已知复数
z=r(cosθ+isinθ)
z^2=r^2(cosθ+isinθ)^2
=r^2(cos^2θ-sin^2θ+isin2θ)
=r^2(cos2θ+isin2θ)
z^3=z*z^2=r(cosθ+isinθ)*r^2(cos2θ+isin2θ)
=r^3(cosθcos2θ+isin2θcosθ+isinθcos2θ-sinθsin2θ)
=r^3(cos3θ+isin3θ)
由此可归纳出
z^n=r^n(cosnθ+isinnθ)
(√3+i)^7
=2^7(√3/2+1/2i)^7
=2^7(cosπ/6+Isinπ/6)^7
=2^7(cos7π/6+isin7π/6)
=2^7(-(√3/2-1/2i)
=-2^6(√3+i)