Step 1:
$(8i)^{\large\frac{1}{3}}=2^{\large\frac{1}{3}}i^{\large\frac{1}{3}}$
$\Rightarrow 2^{\large\frac{1}{3}}[\cos\large\frac{\pi}{2}$$+i\sin\large\frac{\pi}{2}]^{\large\frac{1}{3}}$
$\Rightarrow 2^{\large\frac{1}{3}}[\cos(2k\pi+\large\frac{\pi}{2})$$+i\sin(2k\pi+\large\frac{\pi}{2})]^{\large\frac{1}{3}}$$\;\;k\in z$
$\Rightarrow 2^{\large\frac{1}{3}}[\cos(4k+1)\large\frac{\pi}{2}+$$i\sin(4k+1)\large\frac{\pi}{2}]^{\large\frac{1}{3}}$$\;\;\;k\in z$
$\Rightarrow 2^{\large\frac{1}{3}}[\cos(4k+1)\large\frac{\pi}{6}+$$i\sin(4k+1)\large\frac{\pi}{6}]$$\;\;\;k=0,1,2$
Step 2:
The roots are $2^{\large\frac{1}{3}}(\cos\large\frac{\pi}{6}$$+i\sin\large\frac{\pi}{6}),$$2^{\large\frac{1}{3}}$$(\cos\large\frac{5\pi}{6}+$$i\sin\large\frac{5\pi}{6}),$$2^{\large\frac{1}{3}}$$(\cos\large\frac{3\pi}{2}$$+i\sin\large\frac{3\pi}{2})$
$\Rightarrow 2^{\large\frac{1}{3}}(\large\frac{\sqrt 3}{2}+\frac{i}{2})$$,2^{\large\frac{1}{3}}(\large\frac{-\sqrt 3}{2}+\frac{i}{2})$$,-2^{\large\frac{1}{3}}i$