In each specific example, the shortest way is probably to use the  subsop  command:

restart;	
sol:={v[1]=t,v[2]=3/2*t,v[3]=v[3]}:
subsop(3=(),sol);

R:=1.33*10^(-9)*C/m^2;
evalf[3](op(1,R)*``(`*`(op(2..3,R))));

restart;
expr:=exp(x)^3*sin(x)+3/(exp(x)^n):
A:=exp(t::anything)^n::anything=exp(t*n):
applyrule([1/A,A], expr);

Edit.

restart;
a:= 456;
L:=convert(a,base,10);
Array([seq(L[i],i=-1..-nops(L),-1)]);
# Or shorter
Array(ListTools:-Reverse(L));

Edit.

Use  LinearAlgebra:-Modular  subpackage to work with matrices on a specific finite field.

To generate all such matrices, read the thread  https://mapleprimes.com/questions/211872-Generate-Invertible-4x4-Matrices-Over-F2-

restart;
x, y := r*cos(t), r*sin(t):
plot3d([[x, y, r], [x, y, 0]], r= 0..1, t= 0..Pi, scaling= constrained);

Your equation contains several parameters in addition to variables  x  and  y . For Maple to accept this equation as an ellipse equation, conditions on the parameters are necessary. To get these conditions, we first select complete squares:

restart;
with(geometry):
eqell := expand((x+(1/2)*R1-(1/2)*R)^2/a^2+y^2/b^2-1);
A:=Student:-Precalculus:-CompleteSquare(eqell,x,y);
B, C := selectremove(t->has(t,x)or has(t,y), A);  
Eq:=subsindets(B,`^`,t->applyop(simplify,1,t))=simplify(-C); # Simplified ellipse equation
geometry:-ellipse(ell, Eq, [x, y]) assuming a>0, b>0, a>b; 
detail(ell);

Edit.

Download Ellipse1.mw

Use the appropriate options. An example:

restart;
plot([sin(x),cos(x)], x=0..2*Pi, color=[blue,red], style=point, symbol=[cross,diagonalcross], symbolsize=12, numpoints=50, adaptive=false, size=[800,400], scaling=constrained);

See help on  ?plot,options
If you want to have a solid line as well, and not just some symbols, then use  style=pointline  option.

Remember that  e  is just a Maple symbol (not Euler's number e), and the exponential function  e^x  should be coded as  exp(x) .
First, we plot this function to ensure the existence and uniqueness of the root.
 

Download bisect.mw

acer's way makes sense for a single application. But if this needs to be done several times, then a more significant gain (for arbitrary matrices) is given by applying the procedure:
 

Download Columns.mw

restart;
Eq:=x^2-3=0:
f:=unapply(lhs(Eq), x);
x0:=3.:
for n from 1 do
x||n:=x||(n-1)-f(x||(n-1))/D(f)(x||(n-1));
if abs(x||n-x||(n-1))<10^(-6) then break fi;
od;

sqrt(3.);  # for comparison

When working with expressions with a small denominator (you have  a=10^(-10) ), increased precision is required. You are working with  Digits=10  by default. If you take for example  Digits:=25 , then the difference in results disappears:

Download I_am_lost_new.mw

This is not a continued fraction. You just need to rewrite the integrand as an infinite product  x^(1/2)*x^(1/4)*x^(1/8)* ...

So we get

int(product(x^(1/2^k), k=1..infinity), x);

Another way:
 

Download filledregions.mw

The procedure  P  saves from the list  L  only those pairs of entries  a  and  b  for which  abs(a - b)  equals the minimum distance  m  between the entries of the original list.

Download P.mw