# Thread: simplifying complex numbers

1. ## simplifying complex numbers

1. The problem statement, all variables and given/known data

Simplify $(1+i\sqrt{2})^5-(1-i\sqrt{2})^5$

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{\s qrt{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?

$arccos\frac{\sqrt{3}}{3} \approx 54.74^\circ$

$arcsin\frac{\sqrt{-6}}{3} \approx -54.74^\circ$

2. 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}$

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