1. The problem statement, all variables and given/known data Simplify [itex](1+i\sqrt{2})^5-(1-i\sqrt{2})^5[/itex] 2. Relevant equations \(z=a+bi\) \(z=r(cos\varphi+isin\varphi)\) \(tg\varphi=\frac{b}{a}\) \(r=\sqrt{a^2+b^2}\) 3. The attempt at a solution \((\sqrt{3}(arccos\frac{\sqrt{3}}{3}+iarcsin\frac{\sqrt{6}}{3}))^5-(\sqrt{3}(arccos\frac{\sqrt{3}}{3}+iarcsin\frac{-\sqrt{6}}{3}))^5 \) How will I get integer angle out of here? \(arccos\frac{\sqrt{3}}{3} \approx 54.74^\circ\) \(arcsin\frac{\sqrt{-6}}{3} \approx -54.74^\circ \)
The best thing to do is just to expand the whole thing. \(\(1+\mathrm{i}\sqrt{2}\)^5=1+\mathrm{i}\(5\sqrt{2}\)-20-\mathrm{i}\(20\sqrt{2}\)+20+\mathrm{i}(4\sqrt{2})=1-11sqrt{2}\,\mathrm{i}\) \(\(1-\mathrm{i}\sqrt{2}\)^5=1+\mathrm{i}\(-5\sqrt{2}\)-20-\mathrm{i}\(-20\sqrt{2}\)+20+\mathrm{i}\(-4\sqrt{2}\)=1+11sqrt{2}\,\mathrm{i}\) \(\therefore\ \(1+\mathrm{i}\sqrt{2}\)^5-\(1-\mathrm{i}\sqrt{2}\)^5=-22\sqrt{2}\,\mathrm{i}\)
