Finding the complex square roots of a complex number without a calculator 
The complex number $z$ is given by $z = -1 + (4 \sqrt{3})i$

The question asks you to find the two complex roots of this number in the form $z = a + bi$ where $a$ and $b$ are real and exact without using a calculator.
So far I have attempted to use the pattern $z = (a+bi)^2$, and the subsequent expansion $z = a^2 + 2abi - b^2$. Equating $a^2 - b^2 = -1$, and $2abi = (4\sqrt{3})i$, but have not been able to find $a$ and $b$ through simultaneous equations.
How can I find $a$ and $b$ without a calculator?

Let$$
z^2=(x+yi)^2=−1+4\sqrt3i,
$$
i.e.$$
(x^2-y^2)+2xyi=−1+4\sqrt3i.
$$
Compare real parts and imaginary parts, $$
\begin{cases}
x^2 - y^2 = -1&\qquad\qquad(1)\\
2xy = 4\sqrt3&\qquad\qquad(2)
\end{cases}
$$
Now, consider the modulus: $|z|^2 =|z^2|$, then
$$x^2 + y^2 = \sqrt{\smash[b]{(-1)^2+(4\sqrt3)^2}} = 7\tag3$$
Solving $(1)$ and $(3)$, we get $x^2 = 3\Rightarrow x = \pm\sqrt3$ and $y^2 = 4\Rightarrow y = \pm2$.
From $(2)$, $x$ and $y$ are of same sign,
$$\begin{cases}
x = \sqrt3\\
y = 2
\end{cases}\text{ or }
\begin{cases}
x = -\sqrt3\\
y = -2
\end{cases}
$$
then$$z = \pm(\sqrt3 + 2i).$$
