# How to solve for complex expression?

I'm trying to get the exact answer to the following complex expression, with the approximate value being $$-.12256 + .74486i$$ (according to WolframAlpha). It looks deceptively simple, but I don't think my calculus-level education can get the job done:

$$0\:=z+1-iz^{-\frac{1}{2}}$$

I tried changing it to a different form (this is the form I want to get the answer in), based on Euler's identity:

$$0\:=r\left(-1\right)^{\theta }+1-ir\left(-1\right)^{-\frac{\theta }{2}}$$

or even:

$$0\:=r\left(-1\right)^{\frac{\theta }{\pi }}+1-ir\left(-1\right)^{-\frac{\theta }{2\pi }}$$

Can this be done?

$$z + 1 -iz^{-1/2} = 0$$ $$<=> z + 1 = iz^{-1/2}$$ $$<=> (z + 1)^{2} = (iz^{-1/2})^{2}$$ $$<=> z^{2} + 2z + 1 = -z^{-1}$$ $$<=> z^{2} + 2z + 1 + z^{-1} = 0$$ $$<=> z^{3} + 2z^{2} + z + 1 = 0$$
Solving this equation, you will get three different approximate roots: $$-1.754877...$$, $$−.12256+.74486i$$ and $$−.12256-.74486i$$
About why your form is wrong, let's consider an example with $$z = \frac{5}{2} + \frac{5 \sqrt{3}}{2}i$$.In this example, z has $$r = 5$$ and $$\theta = \frac{\pi}{3}$$. Writing this with the form $$z = re^{i \theta}$$ and using Euler's Identity we get:
$$z = 5e^{i \frac{\pi}{3}}$$ $$= 5(e^{i \pi})^{\frac{1}{3}}$$ $$= 5(-1)^{\frac{1}{3}}$$ $$= -5$$ which contradict to the real value of z above.