exercise authority upon, bring under the (have) power of

Concise Bible Dictionary:

The two principal words in the New Testament translated “power” are 1. δύναμις, and 2. ἐξουσία. It is important to see the difference between them, for their signification is not at all the same. No. 1 may be described as “capacity, moral or physical ability, power.” No. 2 signifies “delegated authority, right, privilege, title.” The latter always supposes power to exercise the right; but in the former there is no thought of right or authority. No. 1 is translated in the AV “ability, might, mighty, mighty deeds, miracles, power, strength, violence, mighty works, wonderful works,” &c. which will help further to show the character of the word, contrasted with No. 2, which is translated “authority, jurisdiction, liberty, power, right, and strength.”
The word “power” occurs in both lists, and this needs to be cleared of any ambiguity. No. 2 is often translated “power” where some other word would convey the sense better; but there is no single word in the English language that exactly answers to the Greek, and which would suit in all places. A concordance must be consulted for a full list of the occurrences: a few passages only are cited. All “authority” is given to the Lord Jesus (Matt. 9:66But that ye may know that the Son of man hath power on earth to forgive sins, (then saith he to the sick of the palsy,) Arise, take up thy bed, and go unto thine house. (Matthew 9:6); Matt. 28:1818And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. (Matthew 28:18); John 17:22As thou hast given him power over all flesh, that he should give eternal life to as many as thou hast given him. (John 17:2)). Satan offered to give to the Lord “authority” over the kingdoms of the world which had been delivered to him, if the Lord would fall down and worship him (Luke 4:66And the devil said unto him, All this power will I give thee, and the glory of them: for that is delivered unto me; and to whomsoever I will I give it. (Luke 4:6)). To as many as received the Lord, to them gave He “right” or “title” to become the children of God (John 1:1212But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name: (John 1:12)). “There is no ‘authority’ but of God.” No. 2 occurring five times in Romans 13:1-31Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God. 2Whosoever therefore resisteth the power, resisteth the ordinance of God: and they that resist shall receive to themselves damnation. 3For rulers are not a terror to good works, but to the evil. Wilt thou then not be afraid of the power? do that which is good, and thou shalt have praise of the same: (Romans 13:1‑3). Along with ‘principality’ occurs No. 2 (Eph. 1:2121Far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come: (Ephesians 1:21); Eph. 3:1010To the intent that now unto the principalities and powers in heavenly places might be known by the church the manifold wisdom of God, (Ephesians 3:10); Eph. 6:1212For we wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places. (Ephesians 6:12); Col. 1:1616For by him were all things created, that are in heaven, and that are in earth, visible and invisible, whether they be thrones, or dominions, or principalities, or powers: all things were created by him, and for him: (Colossians 1:16); Col. 2:10, 1510And ye are complete in him, which is the head of all principality and power: (Colossians 2:10)
15And having spoiled principalities and powers, he made a show of them openly, triumphing over them in it. (Colossians 2:15); Titus 3:11Put them in mind to be subject to principalities and powers, to obey magistrates, to be ready to every good work, (Titus 3:1)).
The principal thing to remember is that No. 2 signifies a delegated right or title, with the presumed power or strength to enforce the right; whereas in No. 1 it is strength or power only.

Strong’s Dictionary of Greek Words:

Number:
1850 (find all occurrences in KJV Bible)
Greek:
ἐξουσιάζω
Transliteration:
exousiazo
Phonic:
ex-oo-see-ad’-zo
Meaning:
from 1849; to control
KJV Usage:
exercise authority upon, bring under the (have) power of