Find all the roots of the complex number $z=8(cos(\frac{\pi }{3})+isin(\frac{\pi }{3}))$.

Step 1 ：Step 1: We write it in exponential form $z=8e^{\frac{i\pi }{3}}$

Step 2 ：Step 2: We know that the roots of the complex number $z^n=r^n(cos(n\theta )+isin(n\theta ))$ are given by $z_k=r^{\frac{1}{n}}(cos(\frac{\theta +2k\pi }{n})+isin(\frac{\theta +2k\pi }{n}))$ where $k=0,1,2,\dots ,n-1$.

Step 3 ：Step 3: We need to find the roots of the equation $z^n=8e^{\frac{i\pi }{3}}$.

Step 4 ：Step 4: For $n=1$, the root is $z_0=8^{\frac{1}{1}}(cos(\frac{\frac{\pi }{3}+2\ast 0\ast \pi }{1})+isin(\frac{\frac{\pi }{3}+2\ast 0\ast \pi }{1}))=8(cos(\frac{\pi }{3})+isin(\frac{\pi }{3}))=8e^{\frac{i\pi }{3}}$.

Step 5 ：Step 5: For $n=2$, the roots are $z_0=8^{\frac{1}{2}}(cos(\frac{\frac{\pi }{3}+2\ast 0\ast \pi }{2})+isin(\frac{\frac{\pi }{3}+2\ast 0\ast \pi }{2}))=2\sqrt{2}(cos(\frac{\pi }{6})+isin(\frac{\pi }{6}))=2\sqrt{2}{e}^{\frac{i\pi }{6}}$ and $z_1=8^{\frac{1}{2}}(cos(\frac{\frac{\pi }{3}+2\ast 1\ast \pi }{2})+isin(\frac{\frac{\pi }{3}+2\ast 1\ast \pi }{2}))=2\sqrt{2}(cos(\frac{5\pi }{6})+isin(\frac{5\pi }{6}))=2\sqrt{2}{e}^{\frac{5i\pi }{6}}$.

Step 6 ：Step 6: For $n=3$, the roots are $z_0=8^{\frac{1}{3}}(cos(\frac{\frac{\pi }{3}+2\ast 0\ast \pi }{3})+isin(\frac{\frac{\pi }{3}+2\ast 0\ast \pi }{3}))=2(cos(\frac{\pi }{9})+isin(\frac{\pi }{9}))=2e^{\frac{i\pi }{9}}$, $z_1=8^{\frac{1}{3}}(cos(\frac{\frac{\pi }{3}+2\ast 1\ast \pi }{3})+isin(\frac{\frac{\pi }{3}+2\ast 1\ast \pi }{3}))=2(cos(\frac{7\pi }{9})+isin(\frac{7\pi }{9}))=2e^{\frac{7i\pi }{9}}$, and $z_2=8^{\frac{1}{3}}(cos(\frac{\frac{\pi }{3}+2\ast 2\ast \pi }{3})+isin(\frac{\frac{\pi }{3}+2\ast 2\ast \pi }{3}))=2(cos(\frac{13\pi }{9})+isin(\frac{13\pi }{9}))=2e^{\frac{13i\pi }{9}}$.