Question 564050
A: Given *[tex \LARGE z = r(\cos(u) + i*\sin(u))] then


*[tex \LARGE z^n = r^n(\cos(u) + i*\sin(u))^n]


By Euler's formula, *[tex \LARGE e^{iu} = \cos(u) + i*\sin(u)] (this is proven using Taylor series -- you'll see these in calculus) so we have


*[tex \LARGE z^n = r^n(e^{iu})^n]


*[tex \LARGE = r^n(e^{i(un)}) = r^n(\cos(nu) + i*\sin(nu))], done.


B: Rewrite using Euler's formula:


*[tex \LARGE x = re^{iu}]


*[tex \LARGE y = te^{iv}]


*[tex \LARGE xy = rte^{iu}e^{iv} = rte^{i(u+v)} = rt(\cos(u+v) + i*\sin(u+v)]