I stick to the first notation of f. For me that is not a function, since RootOf stands
for the set of solutions (hence there is no unique number assigned to n).

One has to select for the roots. And I simplify the results. Then 2 cases remain:

[allvalues(f(n))]: 
simplify(%): evala(%);

h:=unapply(%[1],n);
g:=unapply(%%[2],n);

   h := n -> -1/3*(6*n^2-(21*n^2-24*n+7)^(1/2)-6*n+2)/(2*n^2-3*n+1)

Considering the first gives

[solve(h(n), n)]; 

                               1/2          1/2
                             33           33
                      [1/4 - -----, 1/4 + -----]
                              12           12
map(h,%): simplify(%);

                                [0, 0]


The second possible selection is interesting, it has no solution.

I am using 32 Bit on Win 7

theDLL:="C:\\WINDOWS\\SYSTEM32\\urlmon.dll";

URLDownloadToFile := define_external(
  'URLDownloadToFileA',
  pCaller::integer[4], 
  szURL::string,
  szFileName::string, 
  dwReserved::integer[4], 
  lpfnCB::integer[4],
  'RETURN'::integer[4] ,
  LIB=theDLL):

Url:= " ... "; # as string

myDirectory:= "d:\\temp";
myFile:=      "data.csv";
Destination := cat(myDirectory, "\\", myFile);

DL:= proc() URLDownloadToFile(0, Url, Destination, 0, 0) end proc;
DL(); # now download

-3*b*k*u^2-3*b*k^2*u^4-k^3*u^6*b+k*u+5*k^2*u^3-b;
subs(b=1, k=1, %);
[solve(%, u)]: evalf(%);


  [0.794972852988398 + 0.401160655470270 I,
        -0.102016745838554 + 0.541867900596280 I,
        -0.692956107149844 + 1.91520931667072 I,
        -0.692956107149844 - 1.91520931667072 I,
        -0.102016745838554 - 0.541867900596280 I,
        0.794972852988398 - 0.401160655470270 I]

In my second live I work as privacy officer. And as such I already would stop using
that service at registration: the most idiotic thing I can imagine, that already at
that stage Google analytics is present and tracking.

Similar thoughts for your other data source, which uses Amazon's cloud.

Due to politeness I do not want to comment on Maple :-)

I think it is possible to download such files without browser or Maple in anonymous
way (using WIN DLLs or wget) and that this is a better idea.

In your case actually Maple does it "only" numerically, try it with 32/10 instead of 3.2

Z / nZ = Z / (-n)Z, that's all ...

This is a polynomial of degree 6 and the command 'irreduc' returns true, so it is not likely to be solvable by radical expressions

Setting k=1, b=1 and using fsolve(%, complex) shows, that it may have only non-real roots

The answer is: yes, it has no roots.

That follows by looking at Eigenvalues and Eigenvectors, using linear Algebra only.
The appended sheet shows it and I hope it is correctly written down.

MP_primitive_matri.mws

MP_primitive_matri.pdf

edited: changed file names

You may use "signum".

eq := x = y*(25+2*x)^(2/3)/(25*(.566*(25*x)^(2/3))); 

[seq( eq, y in [1,2,5])];
map(fsolve, %);

    [.205293886670198, .312212133156964, .544930149205805]


Or if you have written it in rational notation you will end up with

RootOf(113325935*_Z^15-256*_Z^6*y^3-6400*_Z^3*y^3-40000*y^3, index=1)^3;
[seq(%, y in [1,2,5])]; 
evalf(%);

    [.205293886670197, .312212133156964, .544930149205804]

why do you expect that? how would you prove it?

sqrt(1+x^2+y^2);
int(%,z=-1..2);                       # does not depend on z, so kill it
Int(%, y=-sqrt(8-x^2)..sqrt(8-x^2));  # and the 2nd integral is 1-dim
Change(%, y=sqrt(8-x^2)*eta, eta);    # get rid of variable bounds
value(%);                             # solve it
Int(%, x=0..2*sqrt(2)); simplify(%);  # set up final task, keep it simple
value(%);                             # now finish it

                                26 Pi

Int(sin(a*x)/x,x=0..infinity);
diff(%, a);

I bet that the (relative) small differences are due to numerical rounding errors (it is unclear, what you are using as computational precision). And some changing ordrer of execution (it is not guaranteed in M that it is always the same)

For uploading your sheet: at the end of the menu of the editor (for contributing to this forum) you find a green arrow. Click it. Another menu opens (like in many forums). Use the first step to locate your file on your local machine, then use the upload command, as it is presented. Now choose "insert as link" (not otherwise).

# alias(beta = RootOf(x^3 + x + 1 = 0));
  alias(beta = RootOf(x^3 + x + 1 ,x));  # I prefer that

  expand(q); evala(%);

  expand(t+s); evala(%);

Then both lines work, giving -beta^2-beta;