I want to understand how I might rearrange an equation with variable exponents to solve for a desired variable.
Take this equation, for example:
$d = \frac{(l(r(1+r)^{m})(s(1+s)^{n}))}{(-r+(1+s)^{n}(r+(-1+(1+r)^{m})s))}$
How can I isolate and solve for $s$?
You can rearrange the equation to be of the form:
$$s(s+1)^{n}[(lr-d)(r+1)^m+d]-rd(s+1)^n+rd=0$$
Which is a polynomial of degree $n+1$ in $s$. You have 6 unknowns including $s$ here.
The types of solutions for $s$ depend on the values for the other unknowns, and the value of $n$ dictates what degree the polynomial is, and how the polynomial could be solved.
Eg the case $n=1$ is much simpler than $n=2,3$ etc.