For general $z\in\mathbb{C}$, the expression $\sqrt{z}$ means the principal square root of $z$.
Assuming that $a,b,c\in\mathbb{R}$ and $c\ge 0$, one can write
$$
w:=a+ bi\sqrt{c}=re^{i\theta}
$$
for some $r\ge 0$ and $\theta\in(-\pi,\pi]$ such that
$$
\sqrt{a^2+b^2c}=r,\quad r\cos\theta = a,\quad r\sin\theta =b\sqrt{c}.
$$
It then follows that
$$
\sqrt{w}=\sqrt{r}e^{i\theta/2}.
$$
Now you use Euler's formula. Finding $r$ is straightforward. One needs the inverse trigonometric functions for $\theta$ in general.
[Added:] If you only consider the general case for $a,b,c\in\mathbb{R}$ and $c\ge 0$, there is nothing special about $\sqrt{c}$ and one can simply consider $w=a+ib$ by introducing a new variable. In this case
$$
r = \sqrt{a^2+b^2},\quad \tan\theta = \frac{b}{a},\quad \theta\in(-\pi,\pi].
$$
[Added later:] In the Wikipedia article mentioned above, one can see the algebraic formula:
When the (complex) number is expressed using Cartesian coordinates the following formula can be used for the principal square root:
$$
{\displaystyle {\sqrt {x+iy}}={\sqrt {\frac {{\sqrt {x^{2}+y^{2}}}+x}{2}}}\pm i{\sqrt {\frac {{\sqrt {x^{2}+y^{2}}}-x}{2}}},}
$$
where the sign of the imaginary part of the root is taken to be the same as the sign of the imaginary part of the original number, or positive when zero. The real part of the principal value is always nonnegative.