Simplify the following and express in the form $a + bi$

Question

**(a)**$(1+\sqrt{3}i)^{4i}$

I have no problem finding the solution as I simply convert the expression $1+\sqrt{3}i$ into euler/polar form and arrive at the following:

let $z=1+\sqrt{3}i$. Then $$z = 2\cdot e^{\frac{\pi}{3}}$$

Then $$z^{4i} = 2^{4i} \cdot e^{i\cdot i \cdot {\frac{4\pi}{3}}}$$

this simplifies to $$z^{4i} = e^\frac{-4\pi}{3}\cdot e^{4i\ln(2)}$$

I'm almost certain my solution is correct, however, I have absolutely no idea how one would convert this back to the form $a+bi$.

Is anyone able to provide a method to do so (or perhaps a different solution that would lead us to the solution in the appropriate form).

Answer 1

You already have your number in polar form $r\,e^{i\theta}$. If $a+ib=r\,e^{i\theta}$, then $$ a=r\cos\theta,\ \ \ b=r\sin\theta. $$ So, in your case, $$ a=e^{-4\pi/3}\,\cos(4\ln2),\ \ \ b=e^{-4\pi/3}\sin(4\ln2). $$

Answer 2

Just use Euler's formula: $e^{ix} = \cos(x) + i \sin(x)$. That's how you go from polar to rectangular.