5. Példa $(1+i\sqrt 3)^9=?$

Megoldás.

a) $(1+i\sqrt3)^9=((1+\sqrt3 i)^4)^2(1+\sqrt3 i)=\left[1+4\cdot\sqrt3 i+6\cdot(\sqrt3 i)^2+4\cdot(\sqrt 3 i)^3+(\sqrt3 i)^4\right]^2(1+\sqrt3 i)$

$=(-8-8\sqrt3 i)^2(1+\sqrt3 i)=64(-2+2\sqrt3 i)(1+\sqrt3 i)=-128\cdot4=-512$

b) $(1+i\sqrt3)^9=\left[2\cdot(\cos({\pi\over3})+i\sin({\pi\over3}))\right]^9=2^9\cdot(\cos({9\pi\over3})+i\sin({9\pi\over3}))=512\cdot(\cos(3\pi)+i\sin(3\pi))=-512$