For the relationship
$$A^2=246*B^5=C$$
When $ C>1$, $$A>B$$ for certain values of C such as $0.25$
$$ A>B$$
$$ A^2=246*B^5=0.25\,$$ $$ A=\sqrt{0.25}=0.5\,$$ $$ B=(0.25/256)^{1/5}=0.25$$
$$\ ∴A>B\,$$
But when C gets closer to $0$ e.g. $3\cdot 10^{-3}$ $$ B>A$$
$$ A^2=246*B^5=3\cdot 10^{-3},$$ $$ A=\sqrt{3\cdot 10^{-3}}=0.05$$ $$ B=(3\cdot 10^{-3}/256)^{1/5}=0.1$$
$$\ ∴B>A\,$$
1.Where is (or how do I find) the turning point between when $A>B$ to $B>A$?
I know it is somewhere at a positive near $0$ value.
2.Why does this occur at such an unusual place? E.g. if $A>B$ for all values over $0$ and $B>A$ for all values <$0$ this would make some sense, but what does it occur at a value between $3E-3$ and $0.25$.
If you are after some context, I ran into the problem when working on a question I asked on Chemistry Stack Exchange here but it is a bit out of scope.
The turning point is when $A=B$, that is $A^2=246A^5$ or $A=\frac{1}{\sqrt[3]{246}}$. Then $$C=A^2=\frac{1}{246^{2/3}}=\frac{\sqrt[3]{246}}{246}.$$