\\|a+ib|=\sqrt{a^2+b^2}\\$But $ a+ib=(x+iy)^n\\\therefore |(x+iy)^n|=\sqrt{a^2+b^2}\\a^2+b^2=|(x+iy)^n|^2=|x+iy|^{2n}=(x^2+y^2)^{2n}\\\\$For the first question, $\\x+iy=(X+iY)^3=X^3 + 3X^2iY - 3XY^2 -iY^3\\$Equating $\Re (z) $ and $ \Im (z),\\x = X^3 - 3XY^2 \ (*)$ and $y = 3X^2Y - Y^3 \ (**)\\...