; A small Mathomatic demonstration.

clear all
y=x_new^n
0 ; solve for zero
taylor x_new 1 x_old ; build the nth root approximation equation
x_new ; solve for the wanted variable
simplify ; convergent nth root approximation equation:
replace x_old x_new with x ; make old and new the same
x ; make sure it is correct by solving for x

y=x^(1/3)+x^(2/3) ; a quadratic, cubed root equation
x ; solve for x
expand
simplify

e^x ; exponential function
taylor x 10 0 ; generate a 10th order taylor series of the exponential function
laplace x ; do a Laplace transform on it
laplace inverse x ; undo the transform