Question

Find the square roots of the complex number

Solution:
z=1 3i

Answer 1

"space" is not a + or -

I'll go with
1+3i = √10 cis 71.56°
so √(1+3i) = ∜10 cis 35.73° = 1.44 + 1.04i

Answer 2

or, if not familiar with De Moivre's Theorem:

let √z = a + bi
(a + bi)^2 = 1 + 3i
a^2 + 2abi + b^2 = 1 + 3i
a^2 - b^2 + 2abi = 1 + 3i

a^2 - b^2 = 1, 2ab = 3 ---> b = 3/(2a)
a^2 - 9/(4a^2) = 1
4a^4 - a^2 - 9 = 0

using the quadratic formulas, and only the real values of a, ..
a = 1.443 or a = -1.443
b = 1.04 or b = -1.04

√z = 1.443 + 1.04i <----- primary root
OR
-1.443 - 1.04i

in De Moivre's method the second square root would be
∜10 cis (35.73° + 180°)
= ∜10 cis 215.73°