How can I draw the complex number $z_1 = 2e^{{\pi\over6}i -1}$

Question

I'm not sure how to draw the following number in the complex plane:

$$z_1 = 2e^{({\pi\over6}i -1)}$$

My guess would be that you seperate the exponents:

$$z_1 = 2e^{({\pi\over6}i)} e^{-1}$$

To get:

$$z_1 = \frac{\sqrt 3}{e} + \frac{1}{e}i$$

So $ \operatorname{Re}(z_1) =\frac{\sqrt 3}{e}$ and $ \operatorname{Im}(z_1) =\frac{1}{e}$.

Is this correct? This is part of a previous exam question, but my teacher doesn't provide any answers.

Answer 1

This is correct. You get the complex number $z$ with modulus $\frac2e$ and argument $\frac{\pi}6$.