<s>[INST] Find the value of $√(4+6\sqrt{5}i) + √(4-6\sqrt{5}i)$ 
Find the value of $\sqrt{(4+6\sqrt{5}i)} + \sqrt{(4-6\sqrt{5}i)}$

$\sqrt{(4+6\sqrt{5}i)} = ± (3+\sqrt{5}i)$
$\sqrt{(4-6\sqrt{5}i)} = ± (3-\sqrt{5}i)$ 
There are two solutions to each which implies on adding them together, we will get four combinations and four different answers.
These answers are $6,-6,-2\sqrt{5}i,2\sqrt{5}i$.
But the only answer given in my book is $6$; they have only taken the + and + combination. Why we don't take the remaining three combinations? Why aren't all four the answers?
 [/INST] I think the book means the following.
$$\left(\sqrt{4+6\sqrt5i}+\sqrt{4-6\sqrt5i}\right)^2=8+2\cdot14=36$$
and the book gets $\sqrt{4+6\sqrt5i}+\sqrt{4-6\sqrt5i}=6$.
This reasoning is wrong, of course.
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