I have one independent variable which has values less than 1, and want to see how the odds of having a disease change when the independent variable increase by 0.1. I ran a binary logistic regression to predict an outcome (disease vs healthy) and as expected it turned out odds ratio had a crazy high value and confidence intervals. Therefore I took the 10th root of odds ratio and CI and now odds are interpretable, however, I don't know if what I did makes sense at all from a mathematical standpoint...
It appears that your issue is that the independent variable has a small range and that, therefore, the odds ratio for a one unit increase are very high.
Rather than take the 10th root (or any root), it is much more straightforward to simply multiply the IV by 10 or 100. This changes nothing about the meaning of the regression, it just makes the output have fewer digits.
This is equivalent to changing units from (say) kilometers to meters.
There may be some situation where taking the 10th root makes sense, but I can't think of one offhand and this certainly doesn't seem like one. But the problem isn't mathematical, it's substantive. What would the 10th root of your IV be?