# expression containing radicals of imaginary numbers

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].

{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 *)

• Thank you very much. Four commands in a row! – imida k Aug 27 '19 at 7:45

Try

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


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

• Thank you very mcuh – imida k Aug 27 '19 at 7:45

Yet another possibility:

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