To find the square root of 5 + 12i, we can use the polar form of complex numbers and De Moivre's theorem.

Step 1: Convert the complex number into polar form.
5 + 12i can be written as r(cosθ + isinθ), where r is the magnitude of the complex number and θ is the argument.

To find r, we use the formula: r = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively.
In this case, a = 5 and b = 12.
So, r = √(5^2 + 12^2) = √(25 + 144) = √169 = 13.

To find θ, we use the formula: θ = tan^(-1)(b/a) = tan^(-1)(12/5).
Using a calculator, we find that θ ≈ 1.176 radians.

Therefore, 5 + 12i can be expressed as 13(cos1.176 + isin1.176) in polar form.

Step 2: Find the square root using De Moivre's theorem.
The square root of a complex number in polar form can be found by taking the square root of the magnitude and halving the argument.

The magnitude of the square root is √r = √13.
The argument of the square root is θ/2 = 1.176/2 = 0.588 radians.

Step 3: Convert the square root back to rectangular form.
Using the polar to rectangular conversion formula, we have:
√13(cos0.588 + isin0.588) = √13(cos0.588) + √13(isin0.588)
= √13(0.843 + 0.537i)
= 0.843√13 + 0.537√13i.

Therefore, the square root of 5 + 12i is 0.843√13 + 0.537√13i.

Comparing this with the given options, we can see that option A, (3 + 2i), is the correct answer.