De Moivre's theorem is most likely the method of solution expected.
You have
Your second expression is incorrect and unnecessary, since is a single point on the complex plane.
However, it does have 2 square roots.
De Moivre's Theorem gives
You will have to use the 2 values of ,......
Alternatively,
Using Pythagoras' theorem, the length of is
Therefore the length of the square root is 2.
The argument of is
To find solutions for from , realise that in squaring , the double angle of is
We therefore also take a 2nd angle for , the angle
These angles are
From an algebraic perspective, has two square roots, and
so the square root of at angle is
The negative square root is
These complex numbers differ by radians.
The reason we take 2 angles is because in the complex world, the angles
and are the same, as they reference the exact same position.
When a complex number is squared, it's angle doubles,
so a complex number with an angle between and
when squared has an angle between and .