I have the following equation, $$z^5 = -16 + (16\sqrt 3)i$$I am asked to write down the 5th roots of unity and find all the roots for the above equation expressing each root in the form $re^{i\theta}$. I am just wondering if my solutions are correct. Here are my solutions,
5th roots of unity, $$z = 1, e^{\frac{2\pi}{5}i}, e^{\frac{4\pi}{5}i}, e^{-\frac{2\pi}{5}i}, e^{-\frac{4\pi}{5}i}$$
and for all the roots for the above equation, $$z = e^{\frac{2\pi k} {5}i}, 2e^{\frac{(6k+2)\pi}{15}i}$$
where k = 0, 1, 2, 3, 4.
Please correct me if there are any mistakes. I will leave my working at the answer section for future reference.
Best Answer
If u have used the theta as 2kπ/theta and got it then its right