But then when I tried doing this by expressing the complex number in polar coordinate form, there are only two roots.

sqrt(3+i) = 10^(1/4) [cos (1/2)tan-1(1/3) + i sin (1/2)tan-1(1/3)]

or 10^(1/4) {cos [(1/2)tan-1(1/3)+pi] + i sin [(1/2)tan-1(1/3)+pi]}

What's wrong?