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Could somebody please verify or help me along with my answers? Thank you.

My updated work(maybe someone can help me a little more now):

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1 Answer

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The first one is incorrect:

$$ e^{i\frac{\pi}{2}} = \cos(\pi/2) + i \sin(\pi/2) \neq 1 + i $$

The second one looks good.

The third and fourth ones you didn't attempt.

Hints: this is how you set these up. If $w = z_1^{z_2}$ where $z_1$ and $z_2$ are complex numbers, then you write $$\log w = z_2 \log z_1$$ so $$w = e^{\log w} = e^{z_2 \log z_1}.$$ (If you just want to take away the shortcut $z_1^{z_2} = e^{z_2 \log z_1}$ I guess that's fine, but I'd be remiss if I didn't at least wave my hands at the "why" of the matter).

For the last one, if $z = re^{i\theta}$, then $$\log z = \log{re^{i\theta}} = \log r + \log{e^{i\theta}} = \dots$$

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