# Finding the roots of complex numbers

$(2i)^{1/2}$

$(1-\sqrt{3}i)^{1/2}$

$(-1)^{1/3}$

$(-16)^{1/4}$

How can I find the roots of the complex numbers above?

Use de Moivre's theorem: $$z^{1/n} = r^{1/n} \exp \left( i\frac{\theta + 2\pi k}{n} \right)$$ for $k = 0, 1, 2, \ldots, n - 1$. The formula is not difficult to derive as we only need to write $z$ in the polar form and take the $n$th root of both sides.
• Here $r$ is the radius, $\theta$ is the angle and $n$ depends on the problem. In general we seek the $n$-th root of a complex number. For example, for ${(2i)^{1/2}}$ we see that $n = 2$ because we want the square root of $2i$. We then use Argand diagram to find $r$ and $\theta$, that is, we represent a complex number as a vector. In this case $r = 2$ and $\theta = \pi /2$ because $2i$ is essentially a horizontal line of length 2. Oct 25, 2012 at 2:10
• In other cases we use the Pythagorean theorem to find $r$, and $\theta$ comes from our knowledge of common angles (usually). Try to find $r$ and $\theta$ for the rest of the complex numbers. Oct 25, 2012 at 2:17
Don't forget that cos and sin functions are $2\pi$ periodic