<s>[INST] How to prove that $\sqrt{-3-2i}+\sqrt{-3+2i} = \sqrt{2(\sqrt{13}-3)}$? Is there a trick to show that
$$\sqrt{-3-2i}+\sqrt{-3+2i} = \sqrt{2(\sqrt{13}-3)}$$ 
is true ?
 [/INST] Just power the left equation like the following:
$$(\sqrt{-3-2i} + \sqrt{-3 + 2i})^2 = -3 - 2i - 3 + 2i + 2\sqrt{(-3 + 2i)\times(-3 - 2i)} = -6 + 2 \sqrt{9 - (2i)^2} = -6 + 2\sqrt{13} = 2(\sqrt{13}-3)$$
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