Conjugates of radicals

Question

I am not sure if one exists but is there a conjugate of the following:

$$\sqrt{3}+\sqrt{2}-\sqrt{5}$$

I attempted it many times but can't get anything.

Answer 1

Depends what you mean by "conjugate". From the point of view of algebraic number theory, this expression has conjugates $$\pm\sqrt3\pm\sqrt2\pm\sqrt5\ ,$$ where any combination of signs is possible. So there are eight answers to your question (one of which is the given number itself).

Answer 2

$$(\sqrt{3}+\sqrt{2}-\sqrt{5})(\sqrt{3}+\sqrt{2}+\sqrt{5})=((\sqrt{3}+\sqrt{2})^2-5)=(3+2-2\sqrt{6}-5)=2\sqrt{6}$$