How to find $\sqrt[3]{8i}$ How do I find the following cube root?
$$\sqrt[3]{8i} = ?$$
I tried by adding $\sqrt[3]{i^3 + 8i + i}$ but that is where my imagination quits.

If you want a general method for finding roots of complex numbers, you may try my answer here, for instance.
In your case, it would work as follows: write your complex number in exponential form,
$$
8i = 8 e^{i\pi/2} \ .
$$
Then, 
$$
\sqrt[3]{8e^{i\pi/2}} = \sqrt[3]{8} e^{i(\pi/2 + 2k\pi)/3} \qquad \text{for} \quad k=0,1,2 \ .
$$ 
That is,
$$
\sqrt[3]{8e^{i\pi/2}} = 2 e^{i(\pi/6 + 2k\pi/3)} \qquad \text{for} \quad k=0,1,2
$$
And this gives:
$$
2e^{i\pi/6} = 2\left(\cos\frac{\pi}{6} + i \sin\frac{\pi}{6}\right) = 2\left(\frac{\sqrt{3}}{2} + i\frac{1}{2} \right) \ ,
$$
$$
2e^{i5\pi/6} = 2\left(\cos\frac{5\pi}{6} + i \sin\frac{5\pi}{6}\right) = 2\left(\frac{\sqrt{3}}{2} - i\frac{1}{2} \right) \ ,
$$
and
$$
2e^{i3\pi/2} = 2\left(\cos\frac{3\pi}{2} + i \sin\frac{3\pi}{2}\right) = -2i \ .
$$
