I have a problem which have happened to me before. It just takes so long between the incidents in different courses I take so I never remember how to approach the issue. I really need some way to think about how to approach this that I might actually remember.
How do I solve for a variable when the terms of it have different exponents?
In my case:
$$\frac{T}{w} = h + \frac{X-1}{h^i}$$
I need to solve for $h$. I just can't find a way to do it.
Any advice would be much appreciated!
You can re arrange this, to get $h^{i+1} - \left(\frac{T}{w}\right)h^i + (X-1)=0$.
Edit: the following only applies if $i$ is a non negative integer.
This is a degree $i+1$ polynomial. This has closed form solutions if $i\leq3$, by the quadratic, cubic or quartic formula. However, it is a well known result of Abel, often proved via Galois theory, that there is no general formula for polynomials of degree 5 or larger.
So, if $i \geq 4$, then you may be lucky and have a closed form solutions if the coefficients happen to work nicely. However, this will not be true in general!
You could try factoring out $h^i$ which would give you $h^i(h - \frac{T}{w})= 1 - X$. This may or may not help, depending on the context.