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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?

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Use e.g. Root[a^3 #1^2 + a^2 #1^4 - 22 a #1^6 + #1^8 &, 5] // ToRadicals. However in general one cannot express Root objects in terms of radicals since it is mathematically impossible. See e.g. How do I work with Root objects? –  Artes Jul 12 '13 at 11:20

1 Answer 1

up vote 2 down vote accepted

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
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Thank you! One more question, for which variable does #1 in this case stand? –  bonanza Jul 12 '13 at 11:26
    
It is an abstract variable see e.g. Slot. Root[a^3 #1^2 + a^2 #1^4 - 22 a #1^6 + #1^8 &, 5] denotes 5-th root of the polynomial a^3 x^2 + a^2 x^4 - 22 a x^6 + x^8, you can get it with e.g. a^3 #1^2 + a^2 #1^4 - 22 a #1^6 + #1^8 &[x] –  Artes Jul 12 '13 at 11:28

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