free, justify(-ier), be righteous

Boyd’s Bible Dictionary:

(judicial righteousness). Through faith in the work of Christ, God places the believer in the state of being permanently righteous before Him (Rom. 5: 16). By the death of Christ, God now offers the free “gift of righteousness” (Rom. 5:1717For if by one man's offence death reigned by one; much more they which receive abundance of grace and of the gift of righteousness shall reign in life by one, Jesus Christ.) (Romans 5:17)).

“Justification” From Concise Bible Dictionary:

The word δικαίωσις occurs but twice in the New Testament, namely, Romans 4:2525Who was delivered for our offences, and was raised again for our justification. (Romans 4:25) and Romans 5:1818Therefore as by the offence of one judgment came upon all men to condemnation; even so by the righteousness of one the free gift came upon all men unto justification of life. (Romans 5:18). In the former passage it appears to be the equivalent in meaning of faith being imputed to the believer for righteousness, that is, of the believer being accounted righteous. Hence the word “justification” may be said to be the estimation formed in God’s mind of the believer in view of that order of things of which Christ risen is the Head. Such estimation has its expression in Christ Himself, and its consequences are seen in Romans 5.
The question as to how a righteous God can justify a sinner is raised and answered in Romans 3. It is difficult to conceive a subject more momentous for every human being. What is set forth in the gospel at the outset is the vindication of God in righteousness as regards sin by the death of the Lord Jesus Christ, where, in God’s infinite grace to sinners, the question of sin and its judgment has been raised between Himself and the spotless Sin-bearer and settled to His glory. Of Him it is said, “Whom God hath set forth a mercy-seat, through faith in His blood....for the showing forth of His righteousness in the present time, so that He should be just, and justify Him that is of the faith of Jesus.” It is then in the blood of Jesus that God’s judgment of sin is seen, and it is on this righteous basis that He can justify all who believe in Him.
Justification of life (Rom. 5:1818Therefore as by the offence of one judgment came upon all men to condemnation; even so by the righteousness of one the free gift came upon all men unto justification of life. (Romans 5:18)) is the righteous bearing into life which is toward all through the one accomplished righteousness of the Lord Jesus Christ even to death, in contrast with the bearing of the one offense of Adam which brought in death and condemnation upon all. What has been effected by the one Man, Jesus Christ, abounds in the scope of it, over all that has been brought in by the one man Adam. In the death of Christ there is seen the complete judgment and removal out of the sight of God both of the sins and of the man who sinned, believers having, through the Lord Jesus Christ raised from the dead, a new Head, in whom they live for God.
There is another aspect of justification referred to in the Epistle of James (James 2), where it is entirely a question of what appears before men. “Show me thy faith without thy works, and I will show thee my faith by my works.”

“Righteousness” From Concise Bible Dictionary:

A term frequently occurring in scripture expressing an attribute of God which maintains what is consistent with His own character, and necessarily judges what is opposed to it—sin. In man also it is the opposite of lawlessness or sin (1 John 3:4-74Whosoever committeth sin transgresseth also the law: for sin is the transgression of the law. 5And ye know that he was manifested to take away our sins; and in him is no sin. 6Whosoever abideth in him sinneth not: whosoever sinneth hath not seen him, neither known him. 7Little children, let no man deceive you: he that doeth righteousness is righteous, even as he is righteous. (1 John 3:4‑7)); but it is plainly declared of man that, apart from a work of grace in him, “there is none righteous, no, not one” (Psalm 14:1-31<<To the chief Musician, A Psalm of David.>> The fool hath said in his heart, There is no God. They are corrupt, they have done abominable works, there is none that doeth good. 2The Lord looked down from heaven upon the children of men, to see if there were any that did understand, and seek God. 3They are all gone aside, they are all together become filthy: there is none that doeth good, no, not one. (Psalm 14:1‑3); Rom. 3:1010As it is written, There is none righteous, no, not one: (Romans 3:10)). But God has, independently of man, revealed His righteousness in the complete judgment and setting aside of sin, and of the state with which, in man, sin was connected. This was effected by the Son of God becoming man and taking on the cross, vicariously, the place of man as under the curse of the law, and in His being made sin and glorifying God in bearing the judgment of sin. Hence grace is established on the foundation of righteousness. The righteousness of God, declared and expressed in the saints in Christ, is thus the divinely given answer to Christ having been made sin. On the other hand, the lake of fire is an eternal expression of God’s righteous judgment. At the present moment God’s righteousness is revealed in the gospel and apprehended by faith.
This is an entirely different principle from that on which the Jew went, namely, that of seeking to establish their own righteousness, and not submitting to the righteousness of God (Rom. 10:33For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God. (Romans 10:3)). Their father Abraham believed God, and it was counted unto him for righteousness; and the faith of the believer is counted to him for righteousness, apart from works (Rom. 4:3,53For what saith the scripture? Abraham believed God, and it was counted unto him for righteousness. (Romans 4:3)
5But to him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness. (Romans 4:5)
).
Christ Jesus is made unto us righteousness from God (1 Cor. 1:3030But of him are ye in Christ Jesus, who of God is made unto us wisdom, and righteousness, and sanctification, and redemption: (1 Corinthians 1:30)). He is the end of the law for righteousness to all those who believe.
Besides the above, there is the practical righteousness which characterizes every Christian. By knowing God’s righteousness he becomes the servant of righteousness. The bride of the Lamb is represented as “arrayed in fine linen, clean and white:” which is “the righteousnesses of the saints” (Rev. 19:88And to her was granted that she should be arrayed in fine linen, clean and white: for the fine linen is the righteousness of saints. (Revelation 19:8)).
The doctrine of the imputed righteousness of Christ, though largely acknowledged in Christendom, is not found in scripture. The explanation generally given of the doctrine is that Christ having perfectly kept the law, His obedience has formed a legal righteousness that is imputed to the believer as if the latter had himself kept the law. One passage of scripture proves this view to be incorrect: “If righteousness come by the law, then Christ is dead in vain” (Gal. 2:2121I do not frustrate the grace of God: for if righteousness come by the law, then Christ is dead in vain. (Galatians 2:21)). The force of the doctrine is to maintain the validity of the law in application to believers; and it stands in the way of their apprehending their death to the law by the body of Christ, so as to be married to Christ raised up from the dead, to bring forth fruit to God (Rom. 7:44Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God. (Romans 7:4)).

Strong’s Dictionary of Greek Words:

Greek:
δικαιόω
Transliteration:
dikaioo
Phonic:
dik-ah-yo’-o
Meaning:
from 1342; to render (i.e. show or regard as) just or innocent
KJV Usage:
free, justify(-ier), be righteous