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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 $s$?

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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.

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    How would your rearrangement look for m = 3 and n = 33, then? Is n = 33 going to be a nightmare? – 2017-02-03
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    Try it and see :-D. You may want to use some of the online solvers.. :) – 2017-02-03