I need to narrow the range (minimum/maximum) to search for a root given the base and the power, the values are all integers, the base can be very large relative to an always positive power. Is there a quick way to determine the order of the root using the relative sizes and values of the base and the power?
E.g.: what would the order of the 11th root of 34567887654323456780987634567895 be?
This answer is a bit too complex for my needs:
Use the number of digits. For your example
I would count that there are $32$ digits, so the radicand can be bounded as follows: $$\left(10^2\right)^{11} = 10^{22} < 34567887654323456780987634567895 < 10^{33}=\left(10^3\right)^{11}$$
That means
$$ 100 < \sqrt[11]{34567887654323456780987634567895} < 1000$$
In general, if your index is $n$ and the radicand is $k$, and the number of digits of $k$ is $d(k)$, then the following are bounds on the "order" of the $n^\text{th}$ root of $k$:
$$\left\lfloor \frac{d(k)-1}{n} \right\rfloor \leq \log_{10}\left(\sqrt[n]{k}\right) \leq \left\lfloor\frac{d(k)-1}{n}\right\rfloor + 1 $$
This uses the floor and ceiling functions, and the logarithm base $10$.