The number $-1$ has no real square roots because there are no real numbers that when multiplied by themselves give a negative product. The solution to this barrier is to define the imaginary number $i$ as the square root of $-1$. By using $i$, it can be shown that you can find general $n$th roots of all numbers, including the complex numbers:

An $n$th root of the number $z$ is a number that when multiplied by itself $n$ times yields $z$.

To find the $n$th roots of a complex number $z$, it is smart to write $z$ in polar form. If you write $z=r{e}^{i𝜃}$, you can easily find a number $w_0$ which fulfills the definition of $n$th root by using normal power rules:

$$w_0={\left(r{e}^{i𝜃}\right)}^{\frac{1}{n}}=r^{\frac{1}{n}}{e}^{i\frac{𝜃}{n}}.$$

To find an $n$th root of $z$, you take the $n$th root of the norm of $z$ and divide the argument of $z$ by $n$. It is also possible to find other $n$th roots of $z$, because the polar form of $z$ is not unique.

A rotation of $2\pi$ radians does not change the value of $z$, so you can write

$$z=r{e}^{i\left(𝜃+2\pi \cdot k\right)}$$

for an arbitrary integer $k\in \mathbb{Z}$. By using the power rules, you can get a general expression for the $n$th roots of $z$:

$$w_k=r^{\frac{1}{n}}{e}^{i\left(\frac{𝜃}{n}+\frac{2\pi }{n}\cdot k\right)}.$$

Every integer $k$ gives you a new $n$th root $w_k$, as long as $k=0,1,2,\dots ,n-1$. When $k$ is $n$ or greater, you will no longer get new $n$th roots. This is because you now have rotated $2\pi$ radians and are back at the starting point $w_0$.

Every $n$th root of $z$ has the same norm, and the difference in argument between $w_k$ and $w_{k-1}$ is $\frac{2\pi }{n}$. If you know $w_k$, you can find $w_{k+1}$ by

$$w_{k+1}=w_{+}\cdot w_k,$$

where $w_{+}=e^{i\frac{2\pi }{n}}$. This is because a multiplication with $w_{+}$ can be thought of as a rotation of $\frac{2\pi }{n}$ radians in the complex plane.

The strategy to find the $n$th roots of $z$ is to first find $w_0=z^{\frac{1}{n}}$ and then the other $n-1$ roots by multiplying by $w_{+}$.

Since all $n$th roots of a number have the same norm, the $n$th roots $w_0,w_1,w_2,\dots ,w_{n-1}$ are uniformly distributed around a circle in the complex plane. The fourth roots of $81$ from Example 1 lie on a circle in the complex plane with radius $3$, and the angle between the roots is $\frac{\pi }{2}$ radians: