Tricky algebra question: Solving $R = \frac{a(SR)^d}{b^d + (SR)^d}$ for $R$

Question

I am trying to solve the following equation for $R$:

$$R = \frac{a(SR)^d}{b^d + (SR)^d}$$

So far I can only find a solution when $d=1$.

Would appreciate any input or help.

Thanks.

Answer 1

For rational $d$,

$$b^d-aS^dR^{d-1}+S^dR^d=0$$

is a polynomial equation or can be made so (by $R^{p/q}=(\sqrt[q]R)^p$).

Such equations are known to have no general closed-form solution, except for a few exponents.

Answer 2

You can write $$Rb^d+S^dR^{d+1}=aS^dR^d$$ and then you will need a numerical method. Remark: $d=2$ and $d=3$ are solvable by radicals. For $d=2$ is one solution: $$R={\frac {1/2\,aS+1/2\,\sqrt {{a}^{2}{S}^{2}-4\,{b}^{2}}}{S}}$$