From: maple@rogers.com   
      
   On Tuesday, October 8, 2019 at 1:34:27 PM UTC-4, clicl...@freenet.de wrote:   
   > acer schrieb:   
   > >   
   > > On Sunday, October 6, 2019 at 10:22:42 PM UTC-4, acer wrote:   
   > > > On Saturday, October 5, 2019 at 11:58:59 AM UTC-4, clicl...@freenet.de   
   wrote:   
   > > > > Hello everybody,   
   > > > >   
   > > > > can your favorite CAS simplify this sum of nested roots   
   > > > >   
   > > > > 5*SQRT(88*2^(2/3) - 145*2^(1/3) + 43)   
   > > > >  + SQRT(97*2^(2/3) - 62*2^(1/3) - 74)   
   > > > >  - 2*SQRT(- 31*2^(2/3) - 37*2^(1/3) + 97)   
   > > > >  - 4*SQRT(6*2^(2/3) + 2*2^(1/3) - 12)   
   > > > >  + 8*SQRT(2^(2/3) - 6*2^(1/3) + 6)   
   > > > >  - 3*SQRT(- 3*2^(2/3) + 3*2^(1/3) + 1)   
   > > > >  - 2*SQRT(- 2*2^(2/3) + 2*2^(1/3) + 4)   
   > > > >  + 6*SQRT(2^(2/3) + 2*2^(1/3) - 2)   
   > > > >   
   > > > > substantially?   
   > > > >   
   > > > > Martin.   
   > > > >   
   > > > > PS: Yes, I know a simple result.   
   > > >   
   > > >   
   > > > Using Maple 2019.1,   
   > > >   
   > > > ee:=5*sqrt(88*2^(2/3) - 145*2^(1/3) + 43)   
   > > >  + sqrt(97*2^(2/3) - 62*2^(1/3) - 74)   
   > > >  - 2*sqrt(- 31*2^(2/3) - 37*2^(1/3) + 97)   
   > > >  - 4*sqrt(6*2^(2/3) + 2*2^(1/3) - 12)   
   > > >  + 8*sqrt(2^(2/3) - 6*2^(1/3) + 6)   
   > > >  - 3*sqrt(- 3*2^(2/3) + 3*2^(1/3) + 1)   
   > > >  - 2*sqrt(- 2*2^(2/3) + 2*2^(1/3) + 4)   
   > > >  + 6*sqrt(2^(2/3) + 2*2^(1/3) - 2):   
   > > >   
   > > > sqrt(evala(ee^2));   
   > > >   
   > > >       4*(1+2^(2/3)-2^(1/3))^(1/2)   
   > >   
   > > Should we take that as "simpler" than,   
   > >   
   > >     4*(-3+3*2^(2/3))^(1/4)   
   >   
   > I suppose the trick of simplifying the squared sum will fail if any one   
   > summand is removed from it, whereas Mathematica's FullSimplify could   
   > then still succeed.   
   >   
   > Martin.   
      
   You have not yet told us what was the form you had previously obtained.   
      
   Aside from that, which do you think is simpler (and why, if it's not just   
   based on leaf-count):   
     4*(-3+3*2^(2/3))^(1/4)   
   or,   
     4*(1+2^(2/3)-2^(1/3))^(1/2)   
      
   Maple 2019.1 returns the following as the result for querying for a   
   algebraic/radical normal form. Both these calls produce this:   
      
   evala(ee):   
   radnormal(ee):   
   lprint(%);   
      
   8*2^(2/3)*(-3*2^(2/3)+3*2^(1/3)+1)^(1/2)+8*2^(1/3)*(-3*2^(2/3)+3*2^(1/3)+1)^(1/   
   2)+12*(-3*2^(2/3)+3*2^(1/3)+1)^(1/2)   
      
   Performing (only) arithmetic simplification of that produces the following:   
      
   nm := simplify(evala(ee),size):   
   lprint(nm);   
      
   (8*2^(2/3)+8*2^(1/3)+12)*(-3*2^(2/3)+3*2^(1/3)+1)^(1/2)   
      
   I noticed that Maple would refuse to "bring" the outer term into the radical,   
   even though its `is` command is capable of ascertaining that the term is real   
   and positive. That's why I took the sqrt of the simplified square.   
      
   Maple recognizes that these various forms are equivalent.   
      
   simplify(sug-ee), simplify(sug-nm), simplify(nm-ee);   
      
                  0, 0, 0   
      
   Of course, if some of the addends in the expression `ee` are removed then one   
   would once again need to check the sign (using signum or csgn if appropriate).   
      
   ff := subsop(8=0, ee):   
   lprint(ff);   
      
   5*(88*2^(2/3)-145*2^(1/3)+43)^(1/2)+(97*2^(2/3)-62*2^(1/3)-74)^(1/2)-2*(-31*2^(   
   2/3)-37*2^(1/3)+97)^(1/2)-4*(6*2^(2/3)+2*2^(1/3)-12)^(1/2)+8*(2^(2/3)-6*2^(1/3)   
   +6)^(1/2)-3*(-3*2^(2/3)+3*2^(1/3)+1)^(1/2)-2*(-2*2^(2/3)+2*2^(1/3)+4)^(1/2)   
      
   sug2 := signum(ff)*simplify(sqrt(evala(ff^2))):   
   lprint(%);   
      
   -2*(-38+25*2^(2/3)+2*2^(1/3))^(1/2)   
      
   nm2 := simplify(evala(ff)):   
   lprint(%);   
      
   (-4*2^(2/3)-10*2^(1/3)-12)*(-3*2^(2/3)+3*2^(1/3)+1)^(1/2)   
      
   simplify(sug2-ff), simplify(sug2-nm2), simplify(nm2-ff);   
      
                  0, 0, 0   
      
   Here's the removal of just the first addend of the original expression `ee`.   
      
   gg := subsop(1=0, ee):   
   lprint(gg);   
      
   (97*2^(2/3)-62*2^(1/3)-74)^(1/2)-2*(-31*2^(2/3)-37*2^(1/3)+97)^(1/2)-4*(6*2^(2/   
   3)+2*2^(1/3)-12)^(1/2)+8*(2^(2/3)-6*2^(1/3)+6)^(1/2)-3*(-3*2^(2/3)+3*2^(1/3)+1)   
   ^(1/2)-2*(-2*2^(2/3)+2*2^(1/3)+4)^(1/2)+6*(2^(2/3)+2*2^(1/3)-2)^(1/2)   
      
   sug3 := signum(gg)*simplify(sqrt(evala(gg^2))):   
   lprint(%);   
      
   (891+1896*2^(2/3)-3081*2^(1/3))^(1/2)   
      
   nm3 := simplify(evala(gg)):   
   lprint(%);   
      
   (23*2^(1/3)+7-2*2^(2/3))*(-3*2^(2/3)+3*2^(1/3)+1)^(1/2)   
      
   What are the criteria for simplification here? Utility for subsequent   
   computation as a normal form (where zero-recognition is desirable, of course)?   
   Leaf count? Oder...?   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)