(* 1° *)

expr=Sum[1/(k+x),{k,5}]

expr=Together[expr]

Apart[expr,x]

(* 2° *)

sol=Solve[{(x^2)+y==c,x+b y==0},{x,y}]

expr={(x^2)+y==c,x+b y==0}/.sol

Simplify[expr]

(* 3° *)

f=Sqrt[1+Exp[x]]

int=Integrate[f,x]

t=D[int,x]

t=D[int,x]

Simplify[t]

(*  4°  *)

m=(b IdentityMatrix[4]-{{0,1,0,0},{0,0,1,0},{0,0,0,1},{1,0,0,0}});

eq=Thread[m.{x1,x2,x3,x4}=={1,0,0,0}]

Solve[{b x1-x2==1,b x2-x3==0,b x3-x4==0,-x1+b x4==0},{x1,x2,x3,x4}]

LinearSolve[m,{1,0,0,0}]

(* 5°  *)

expr=Sqrt[x+y]/(1+Sqrt[x+y])+(1-Sqrt[x+y])/(1-x-y)

Together[expr]