Converting to complex form Write each of the given numbers in the form $a+bi$.
a.) $e^{\frac{-i\pi}{4}}$
b.) ${\frac{e^{1+i3\pi}}{e^{-1+i\pi /2}}}$
c.) $e^{e^i}$
For a, I got $(\frac{\sqrt 2}{2} -\frac{\sqrt 2}{2}i)$, which I think is right. I am stuck on how to do b and c.

You are right on $a$.
For $(b)$, $$\dfrac{e^{1+3 \pi i}}{e^{-1 + i \pi/2}} = \underbrace{e^{2+ 5 \pi i/2} = e^2 e^{i \pi/2}}_{\text{since }e^{2\pi i+ i \theta} = e^{i \theta}} = e^2 i$$
For $(c)$, note that $e^{i} = \cos(1) + i \sin(1)$.
$$e^{e^i} = e^{\cos(1) + i\sin(1)} = e^{\cos(1)}  e^{i\sin(1)} = e^{\cos(1)}\left(\cos(\sin(1)) + i \sin(\sin(1))\right)$$
