syntax - How to get 'exact' expression for result

I have a result from Mathematica for which want to have the exact expression, i.e. all of the arguments of type # replaced. It is given by:

{ a > 0, 
  Inequality[ 0, Less, r, LessEqual, Root[a^3 #1^2 + a^2 #1^4 - 22 a #1^6 + #1^8 & , 5]]}

where a and r are the only two variables. What exactly does the #1 mean here? To which variables does it refer? How do I make Mathematica replace all these place holders with its corresponding variable?

Answer

Root objects are just exact expressions, if possible one can express them in terms of radicals (e.g. with ToRadicals). To deal with inequalities the best approach might be Reduce:

ToRadicals @ Reduce[{ a > 0, 

                      Inequality[0, Less, r, LessEqual, 
                      Root[a^3 #1^2 + a^2 #1^4 - 22 a #1^6 + #1^8 &, 5]]},
                      a] // ComplexExpand

 r > 0 && a >= -(r^2/3) + 2/3 Sqrt[67] r^2 Sin[1/3 ArcTan[227/(3 Sqrt[127947])]]

% // TraditionalForm

enter image description here

and we can get an arbitrarily precise numerical approximation in a standard way, e.g. 40-digit precission:

%% // N[#, 40] &

r > 0 && a >= 0.04555316444285376876486224542203853971094 r^2

Comments

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