solve command, which is part of the core of Maple, is one of the main and frequently used commands. It is unfortunate that it can not cope with such a simple equation. I think that developers should deal with this situation.
We can still solve this equation with solve command in two steps, first solving for  sin(t) :

solve(exp(2*sin(t))-1=0, sin(t));
solve({sin(t)=%, t>=0, t<=16}, allsolutions, explicit);
                                                            0
                    {t = 0}, {t = Pi}, {t = 2 Pi}, {t = 3 Pi}, {t = 4 Pi}, {t = 5 Pi}
 

If you do not know which command to use to finally simplify your numerical expression, it is useful to first calculate it approximately accurately (by  evalf  command), and then, for complete certainty, select the appropriate command for symbolic simplification, in fact already knowing the result.

S:=-1/2*((-30*sqrt(3)+81*I)^(2/3)+21+2*sqrt(3)*(-30*sqrt(3)+81*I)^(1/3)-6*(-30*sqrt(3)+81*I)^(1/3)+I*sqrt(3)*(-30*sqrt(3)+81*I)^(2/3)-21*I*sqrt(3))/(-30*sqrt(3)+81*I)^(1/3);
evalf[20](S);
factor(S);  # or
radnormal(S);
 

Your set is just one octant in 3d (in fact, all points in the space with negative coordinates) .

Download PS.mw

In this example, we generated an arbitrary polynomial of the third degree in 5 variables. But the method itself is of a general nature and it can be used to generate a polynomial of any degree and any number of variables.

polynom.mw

Edit.

Here is a simple procedure that solves the problem for not too large  d  and  K :

Pell:=proc(d::posint,K::posint)
local k, y, x, L;
k:=0; L:=table();
for y from 1 to K do
x:=isqrt(d*y^2+1);
if x^2-d*y^2=1 then k:=k+1; L[k]:=[x,y] fi;
od;
convert(L,list);
end proc:

Example of use:

Pell(27,1000000);

Output:
                  [[26, 5], [1351, 260], [70226, 13515], [3650401, 702520]]

Addition. Of course, much more efficient code can be written if we use  isolve  command, but I think that for a beginner the above will be enough.

Edit.

Specify the range for  x  variable explicitly:

with(DocumentTools):
A := [plot(sin(x)/x, x = -2*Pi .. 2*Pi, legend = sinc(x)), Sum(f(i), i = 1 .. N), plot3d(y^2*sin(x), x = -Pi .. Pi, y = -2 .. 2)]:
Tabulate(A, width = 60):

 

Download Tab.mw

Maximize (or Minimize) command only guarantees a local extremum, not a global one (see help for details). To get a global maximum, first plot a graph, and then narrow the range for the variable  x , that contains the maximum:

Addition. In order to immediately get the global extremum, you can use DirectSearch  package, which should be downloaded from the Maple Application Center and installed on your computer

https://www.maplesoft.com/applications/view.aspx?sid=101333

Download Max.mw

First you need to open your file, then  ctrl+ p

Another way: menu file -> print  or  print preview

Probably a double for loop is meant for summing such terms, if  i  runs from  0  to  m  and  j  runs from  0  to  n . A simple procedure named  Bxy  solves the problem:

Bxy:=proc(m::nonnegint,n::nonnegint)
local S, i, j;
S:=0;
for i from 0 to m do
for j from 0 to n do
S:=S+binomial(m,j)*binomial(n,i)*x^i*(1-x)^(m-i)*y^j*(1-y)^(n-j);
od; od;
S;
end proc:

Example of use:

Bxy(3,2);

                  

Here is the numerical solution:

restart;
pde:=diff(u(x,t), t$2)+diff(u(x,t), x$4)=0;
bc:=u(0,t)=-12*t^2, u(1,t)=1-12*t^2, D[1,1](u)(0,t)=0, D[1,1](u)(1,t)=12;
ic:=u(x,0)=x^4, D[2](u)(x,0)=0;
sol:=pdsolve(pde, {ic,bc}, u(x,t), numeric);
sol:-plot3d(t = 0 .. 1, x = 0 .. 1, axes = boxed);

Use  %.  or  %*  instead of  * :

sum(i %. a, i=1..3);
                                               

Another option is to use  ``(...)  construction to avoid the summation of similar terms. Then all the individual terms of the series are enclosed in parentheses, and it is convenient to compare:

sum(``(i*a), i=1..3);
                                               

 This construction is often useful in other cases. For example, if you need to clearly show the definition of the number  10!  then it can be done in 3 ways:

``(10)!=product(``(k), k=1..10);
# or
``(10)!=`%.`(seq(k, k=1..10));
# or
``(10)!=`%*`(seq(k, k=1..10));
                                          

Edit.                                   

Try this:

simplify(-sqrt((r-2*M)*(-r/(-r+2*M))^eta/r)*X*sqrt(-((r-2*M)/r)^eta*r/(-r+2*M))/f) assuming r>0, r-2*M>0, eta::real;

Edit.

In fact, Maple knows this, but this syntax  convert(sin(alpha), radical)  is required for output. Maple can express in real radicals trigonometric functions of any angle multiple of 3 degrees:

Download sine.mw

You can do this using  plots:-textplot  and  plot  commands as follows:

The code:

restart;
with(plots):
T:=[[0,0.9, "2*5+8 = 2*5+8"], font=[times,20]]:
T1:=textplot(op(T)):
Line:=plot([[t,0.87,t=-0.56..-0.28],[t,0.87,t=0.1..0.38]], color=black):
display(T1, Line, size=[300,70], view=[-1..1,0.6..1.3], axes=none);

Output:
                                            

You can even make an animation of this. For the animation we used the technique from this post .

The code:

A:=animate(textplot,subsop(1=[T[1,1..2][],T[1,3][1..round(i)]],T), i=0..length(T[1,3]), frames=40, paraminfo=false, view=[-1..1,0.6..1.3], size=[300,70], axes=none):
B1:=animate(plot,[[t,0.87,t=-0.56..a], color=black], a=-0.56..-0.28, frames=10, paraminfo=false):
B2:=animate(plot,[[t,0.87,t=0.1..a], color=black], a=0.1..0.38, frames=10, paraminfo=false):
display([A, display(op([1,-1,1],A),B1), display(op([1,-1,1],A),op([1,-1,1],B1),B2)], insequence);

  Output: 

Premature calculation occurs, before the substitution of values  i  into the series. This is easy to see if you write down it as an inert sum:

Sum(0^i/i!, i=0..infinity);
                                                      

 
For correct calculation use  eval  command:

eval(sum(x^i/i!, i=0..infinity), x=0);
                                                             1