Oblicz przyklad

[chrisdk]

Przedstaw dzialanie \(\displaystyle{ \frac{1}{(1-i \sqrt{3})^{3}}}\) w formie \(\displaystyle{ \frac{a}{b}}\), gdzie \(\displaystyle{ a,b \in \mathbb{Z}}\)

[soku11]

\(\displaystyle{ \frac{1}{(1-i\sqrt{3})^3}=
\frac{(1+i\sqrt{3})^3}{(1-i\sqrt{3})^3(1+i\sqrt{3})^3}=
\frac{(1+i\sqrt{3})^3}{4^3}=
\frac{[2(\frac{1}{2}+i\frac{\sqrt{3}}{2})]^3}{64}=
\frac{2^3(\frac{1}{2}+i\frac{\sqrt{3}}{2})^3}{64}=
\frac{(\frac{1}{2}+i\frac{\sqrt{3}}{2})^3}{8}=
\frac{(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3})^3}{8}=
\frac{\cos\pi+i\sin \pi}{8}=
\frac{-1}{8}}\)

Pozdrawiam.