Radiationless energy transfer: how to arrive at the equation that give R0 between two chromophores? I have a homework question where I should calculate  $R_0$ of an energy transfer between two chromophores. However, I don't understand how my professor arrived at the equation that we should apply (which is important to understand for the exam).


  In a radiationless energy transfer experiment (RET) it was observed that the efficiency of transfer was $0.0185$ at $\pu{5 nm}$ distance between the two chromophores. Estimate $R_0$ for the two chromophores, assuming the energy transfer efficiency is $0.7$ at short distances.


In our lecture we are given the formula:
$$[E_\mathrm{eff}] = \frac{R_0^6}{r^6 + R_0^6}$$

I understand that $E_\mathrm{eff}$ is the efficiency of the transfer, and $r$ is the non-variable distance between the two chromophores. However, I am not able to solve for $R_0$. I tried to ask my professor but he thinks faster than I do and skips steps. So I was not able to follow completely, but he somehow mathematically solved the equation to (where $Eff_\mathrm{max}$ is maximum efficiency of energy transfer): 
$$r = R_0 \sqrt[6]{\frac{E_\mathrm{eff}}{E_\mathrm{eff,max}}}$$

When I apply his formula I "plug in" the given numbers and solve for $R_0$ as following:
\begin{align}
R_0 &amp;= \frac{r}{\sqrt[6]{\frac{E_\mathrm{eff}}{E_\mathrm{eff,max}}}}\\
R_0 &amp;= \frac{\pu{5 nm}}{\sqrt[6]{\frac{0.0185}{1}}}\\
R_0 &amp;= \pu{9.72 nm}
\end{align}

I would like to understand how he arrived at the final formula that I apply. Is it correct to solve for $R_0$ like I have shown above?
I think I found the answer to my own question through some help from some classmates. What the professor did, was to solve the equation mathematically by converting $R_0^6$ to $1$. 

So first, the efficiency of the transfer $E_\mathrm{eff}$ is actually a ratio of the actual transfer efficiency divided by the maximum efficient (which we should assume is "one", unless we are told otherwise. 

The next step is a bit hard to explain, as I am not that good in math, but I will try my best. What I understand is that you can "gather" the two $R_0^6$ by multiplying both the numerator and denominator by what we should think of as the number one, but we actually divide by $R_0^6$ and in this way we don’t have to include $R_0^6$ that we divide by on both sides of the equal sign - as it is technically the number one (mathematically allowed), as following: 
$$\frac{E_\mathrm{eff}}{E_\mathrm{eff,max}} = \frac{\frac{R_0^6}{R_0^6}}{\frac{R_0^6}{R_0^6}+ \frac{r^6}{R_0^6}}$$

We then get as following:
$$\frac{E_\mathrm{eff}}{E_\mathrm{eff,max}} = \frac{1}{1 + \frac{r^6}{R_0^6}}$$

So by applying some math and assumptions we can solve the equation like this, and what I have explained in my question should therefore be correct.