with(PolynomialIdeals);
R := PolynomialRing([x, y, z, u, v], 'lex');
P := PolynomialIdeal([x^7*v, y^8*v, z^9*v, u*v]);
decomposition := PrimaryDecomposition(P);
htP := nops([decomposition]);

f := x -> x^3:
g := x -> x^(1/3):

#For complex domain these answers are correct:   
f(g(x));
g(f(x));

#For real domain these answers are correct:
A := [x < 0, x >= 0]:
f(g(x)) assuming x, real;
proc(E) global A; local a; local S := proc(E, a) a, simplify(E) assuming a end; 'piecewise'(seq(S(E, a), a = A)); end(g(f(x)));
F2 := %:

plot([Re, Im](F2), x = -2 .. 2, color = [red, blue], axes = framed);
plot([Re, Im](g(f(x))), x = -2 .. 2, color = [red, blue], axes = framed);

After simplifying for real t, res appears as:

-abs(t)^3*csgn(2*sqrt(2)+3*ln(t))+t^3;

If You know the final form then you can demonstrate with this:

R := RootOf(4*_Z^2+(4*RootOf(60*_Z^3-60*_Z^2+15*_Z-1)-4)*_Z+4*RootOf(60*_Z^3-60*_Z^2+15*_Z-1)^2-4*RootOf(60*_Z^3-60*_Z^2+15*_Z-1)+1);
final := 1/3+(1/3)*cos((1/3)*arctan(3/4));

Finalform := (1/3)*cos(alpha)+1/3;
(`@`(combine, factors, evala, Norm))(Finalform-R)[2, 1, 1];
subs(solve(%, {alpha}), Finalform);
evalf(Testzero(evalf(%-final)), 20);

#or more precisely:

Finalform := (1/3)*cos(arctan(alpha)/3)+1/3;
(`@`(combine, factors, evala, Norm))(Finalform-R)[2, 1, 1];
subs(solve(%, {alpha}), Finalform);
evalf(Testzero(evalf(%-final)), 20);

Also additional assumptions should use command additionally not assume. Using assume again on the same variable reinitializes all assumptions on it (and related variables I believe).

ode := diff(y(x), x) = sin(y(x))+1;
ic := y(0) = Pi;
Order := 40;
sol := dsolve([ode, ic], y(x), series);
ser1 := gfun[seriestorec](subs(sol, y(x)), a(n));
diffeq1 := gfun[rectodiffeq](ser1[1], a(n), y(x));
dsol1 := dsolve(diffeq1, y(x));

This should help:

indets(cS(a)(b)(c), specfunc(anything, cS));

Maple does not resolve types in the 0th operand.

Replace
Int(unapply(eq7,t),-25..25,method=_d01ajc)
with
Int(eq7,t=-25..25,method=_d01ajc)

Instead of ifactor use ifactors. Much easier to work with.

y = -1/2 .. 3, z = -3 .. 3, x = 0 .. 1;
u = 2*x*y, v = 2*x*z;
u = (minimize .. maximize)(2*x*y, x = 0 .. 1, y = -1/2 .. 3);
v = (minimize .. maximize)(2*x*z, x = 0 .. 1, z = -3 .. 3);
u = -1 .. 6, v = -6 .. 6, x = 0 .. 1;
uvEQ := collect(combine(subs(solve({u = 2*x*y, v =  2*x*z}, {y, z}), expand([eq1, eq2]*x^2*300))), [exp, sin, cos]);

Digits := 16;
PLT1:=plots[intersectplot](
op(map(E->map(evalf, surface(E, u = -1 .. 6, v = -6 .. 6, x = 0 .. 1)), uvEQ)),
maxlev = 5, maxtet=100000, grid = [31, 61, 11], thickness = 3, transparency = 0.3):
print(PLT1);

subs(solve({u = 2*x*y, v = 2*x*z}, {y, z}), [y, z, x]);
F1 := unapply(%, [u, v, x]);
plots[display](plottools[transform](F1)(PLT1), view=[ -1/2 .. 3,  -3 .. 3,  0 .. 1]);

If you'd like a plot of the solution you can use:

Digits := 16;
plots[intersectplot](
map(evalf, surface(eq1, y = -1/2 .. 3, z = -3 .. 3, x = 0 .. 1)),
map(evalf, surface(eq2, y = -1/2 .. 3, z = -3 .. 3, x = 0 .. 1)),
maxlev = 5, grid = [31, 61, 11], thickness = 3, transparency = 0.3);

You could use this, if you mean exact match.

ArrayTools:-IsZero(A) and LinearAlgebra:-Dimensions(A)=(4,4);

Also see ?ArrayTools:-IsEqual

expand(rel(n+1));

This works:

restart;
expr := -(r0+Delta_r)^2*(46*r0-41*Delta_r)*r0^5;
subsop(2=b,3=c,4=d, 1=a, expr);

Reasoning is obvious. subsop(1=a, expr) replaces the constant of multiplication. Operand 1 (always the constant) stops being a constant, so a new expression with 2 elements is formed: a*(rest of expression). There is no operand 3 and 4. Your subsop worked as intended.

Try this code, if you have the newer Maple version (looks like you're trying to use old Maple).

y:=sqrt((x^2*(x^2-4))/(x^2-1));
solve(evalc(Im(y)), x);
F := [Minimize = minimize, Maximize = maximize];
G := 'simplify(evalc(Re(y)))';
F(`assuming`([G], [-1 <= x, x <= 1]), x = -1 .. 1, location);
F(`assuming`([G], [2 <= x]), x = 2 .. infinity, location);
F(`assuming`([G], [x <= -2]), x = -infinity .. -2, location);