expression containing radicals of imaginary numbers

Question

I can't bear an expression containing radicals of imaginary numbers, in case it can be expressed as in terms of radicals of real numbers only.

For example, I can't bear the expression

Sqrt[2 + I]

because it can be expressed as

Sqrt[1/2 (2 + Sqrt[5])] + Sqrt[1/2 (-2 + Sqrt[5])] I 

But it seems there is no easy way to do it in Mathematica. I have tried many commands (in Mathematica) but all in vain.

Is there a systematic way to do such job ?

Sqrt[2 + I] was a very simple example. I hope the method work for much more complicated expression.

P.S I know that there are many algebraic numbers that cannot be expressed as in terms of radicals, For example, Root[#^5 + # - 1 &, 1].

Answer 1

{1, I}.(Sqrt[2 + I] // ReIm // ComplexExpand // FunctionExpand // 
   FullSimplify)

(* I Sqrt[1/2 (-2 + Sqrt[5])] + Sqrt[1/2 (2 + Sqrt[5])] *)

or

% // Simplify

(* (I Sqrt[-2 + Sqrt[5]] + Sqrt[2 + Sqrt[5]])/Sqrt[2] *)

Verifying that these are equivalent to the original form

Sqrt[2 + I] === (% // FullSimplify) === (%% // FullSimplify)

(* True *)

Answer 2

Try

Sqrt[2 + I] // ComplexExpand // FunctionExpand

It may not create a result simplified in the exact form you want, but it will be closer.

Answer 3

Yet another possibility:

ComplexExpand[Sqrt[2 + I], TargetFunctions -> {Re, Im}] // FunctionExpand // Simplify
   (5^(1/4) ((2 + I) + Sqrt[5]))/Sqrt[10 + 4 Sqrt[5]]