Math Genius: Finding all solutions $z=x+iy$ for four basic problems

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I am preparing for an exam on Complex Analysis and I have four small problems. Two of them I think I have solved but I would really appreciate a sanity check since I very often miss something. On the other two, I have no idea how to start, any hint/suggestion would be appreciated. I need to find all solutions $z=x+iy$.

$textbf{a) } z+1 = log{(i+1)}$. The typesetting won’t allow it, but is is actually Log, hence the principal branch.

$begin{align}
z+1 &= log{i+1}\
z+1 &= ln{sqrt{2}} + frac{pi}{4}\
z &= ln{sqrt{2}} + frac{pi}{4} -1
end{align}
$

$textbf{b) } sin{z} = -i$. My solution is

$begin{align}
frac{e^{iz}-e^{-iz}}{2i} &= -i\
e^{iz}-e^{-iz} &= 2\
e^{iz}-e^{-iz} -2 &= 0 text{ , now let $a = e^{iz}$}\
a^{2}-2a-1 &= 0\
a &= 1 pm sqrt{2}
e^{iz} = 1 pm sqrt{2}\
iz &= ln{(1 pm sqrt{2})}\
z &= -iln{(1 pm sqrt{2})}.
end{align}
$

Now the two problems I do not know where to start:

$textbf{c) } (z+i)^{3} = 8. $ Writing it out does not seem to help me get anywhere, I then find $z^{3}-3z+3iz^{2}-i=8$.

$textbf{d) } (z+1)^{2i} =1$. No clue.

The first two are correct.

c) $(z+i)^3=8iff z+i=2vee z+i=2left(-frac12pmfrac{sqrt3}2iright)$

d) I assume that $z^w$ is defined as $expleft(woperatorname{Log}(z)right)$. If so,begin{align}(z+1)^{2i}=1&iffexpleft(2ioperatorname{Log}(z+1)right)=1\&iff 2ioperatorname{Log}(z+1)=2pi intext{ (for some integer }ntext{)}\&iffoperatorname{Log}(z+1)=pi n\&iff z+1=e^{pi n}.end{align}

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