Conformation of Basic De Moivre Formula Hello I am just wondering if my understanding of De Moivre formula is correct. Say we want to solve $$(1+i)^{100}$$ then can we simply just do $$(1+i)^{100}=\sqrt{2}^{100}((\cos(50\pi)+i\sin(50\pi))$$ Is that correct application? Thanks

This is almost correct. To use De Moivre's formula you need to get the complex number into polar form (which you have clearly attempted to do). In this case we get $r=\sqrt{1^2+1^2}=\sqrt{2}$ and $\theta=\arctan(1/1)=\pi/4$. Hence $1+i=\sqrt{2}e^{i(\pi/4)}$. Then applying De Moivre's theorem we get:

$$ \begin{split} (1+i)^{100}&=(\sqrt{2}e^{i(\pi/4)})^{100} \\\ &=\sqrt{2}^{100}e^{i(25\pi)} \\\ &=\sqrt{2}^{100}(\cos(25\pi)+i\sin(25\pi)) \end{split} $$