Question

Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the form a + bi.)

(3 + √3i)^4

Answer 1

\[(a+bi)^n= re^{i* n* \phi}\]
\[r=\sqrt{3^2+(\sqrt{3})^2}=\sqrt{12}=2\sqrt3; \phi = \arctan(\sqrt{3}/3) \implies \phi \approx 30^0\]
\[\implies (3+\sqrt3i)^4 \approx (2 \sqrt3)^4e^{4 * i * 30} = 144(\cos(120)+\sin(120)i)\]\[=144(-1/2+\sqrt3/2i)=72(-1+\sqrt3i)\]
http://www.wolframalpha.com/input/?i=%283%2Bsqrt%283%29i%29%5E4

Answer 2

That first line should say:
\[r^n e^{i * n * \phi}\]***

Answer 3

thax :)