# nth Roots of Complex Numbers Examples 1

Recall from the nth Roots of Complex Numbers page that if $z = r (\cos \theta + i \sin \theta) \in \mathbb{C}$, $z \neq 0$, then the $n$ many $n^{\mathrm{th}}$ roots of $z$ are given by the following formula for each $k \in \{ 0, 1, ..., n - 1 \}$:

(1)We will now look at some examples of computing the $n^{\mathrm{th}}$ roots of some complex numbers.

## Example 1

**Compute the square roots of $1 + i$.**

Since we want to compute the square roots of $1 + i$ we have that $n = 2$. We have that $r = |z| = \sqrt{1^2 + 1^2} = \sqrt{2} = 2^{1/2}$. Lastly, we have that $\displaystyle{\theta = \mathrm{Arg}(z) = \frac{\pi}{4}}$. Therefore the square roots of $1 + i$ are:

(2)## Example 2

**Compute the cube roots of $1 + i$.**

This is similar to the previous example. Since we want to compute the cube roots of $1 + i$ we have that $n = 3$. We have again that $r = |z| = \sqrt{1^2 + 1^2} = \sqrt{2} = 2^{1/2}$ and that $\displaystyle{\theta = \mathrm{Arg}(z) = \frac{\pi}{4}}$. Therefore the cube roots of $1 + i$ are:

(3)