I liken the meaning of exousia to having a driver’s licence. When you have a driver’s licence you have the authority, right and freedom to drive a vehicle on public roads.
Here is a sample of most of the verses from 1 Corinthians (NABS) of exousia which demonstrates a range of usage. (I have italicised the English translation of exousia.)
The wife doesn’t have authority over her own body, but the husband. Likewise also the husband doesn’t have authority over his own body, but the wife (1 Cor 7:4.) [Compare this translation with the very good NLT translation.]
But he who stands steadfast in his heart, having no necessity, but has power over his own heart, to keep his own virgin, does well (1 Cor 7:36).
But be careful that by no means does this liberty of yours become a stumbling block to the weak (1 Cor 8:9).
Have we no right to eat and to drink? (1 Cor 9:4.)
Have we no right to take along a wife who is a believer, even as the rest of the apostles, and the brothers of the Lord,and Cephas? (1 Cor 9:5.)
Or have only Barnabas and I no right to not work? (1 Cor 9:6.)
If others partake of this right over you, don’t we yet more? Nevertheless we did not use this right but we bear all things, that we may cause no hindrance to the Good News of Christ (1 Co 9:12.)
Then the end comes, when he will deliver up the Kingdom to God, even the Father; when he will have abolished all rule and all authority and power (1 Cor 15:24).
As to what authentein means? That's a whole other ball game.
Most lexicons give the meaning of the verb authenteō as something like "domineer". But authentein is not a verb. Albert Wolters 1 notes that the cognate nouns authentēs and authentia are used in texts about Gnosticism that date from the 1st and 2nd centuries (as well as in other contexts.) But authentein is not a noun. It is in the infinitive.
In the TLG, a data base of all known ancient Greek literature, authentein occurs only 19 times, 15 of those times are references to 1 Timothy 2:12. So we don't have much to go on there. However, I suggest, based on what we do know, that exousia and authentein are very different as far as "shades of meaning" go.
Furthermore, cognates of authentein do occur in the LXX. In Wisdom 12:6 authenta is used describe parents who murder their children. A cognate also occurs in 3 Maccabees 2:29.
It is important to note that the English word "authority" is not etymologically related to "athentein".
I hope this helps.
- Albert Wolters, “A Semantic Study of authentēs and its Derivatives” in the Journal for Biblical Manhood and Womanhood, 1/11, Spring 2006, p44-65.