AND AGAIN PRAISE THE LORD ALL YOU GENTILES: kai patarlin aineite (2PPAM) panta ta ethne:

In quoting (Ps 117:1 - Spurgeon's Note), Paul's main point again is to highlight that the Gentiles had been invited to sing praises to the Lord, the God of Israel in the OT. Thus Paul is emphasizing that the work of the Messiah had always pointed to bringing the Gentiles together into one body with the Jews. The Gentiles were not an afterthought.

AND LET ALL THE PEOPLES PRAISE HIM: kai epainesatosan (3PAAM) auton pantes hoi laoi:

AND AGAIN ISAIAH SAYS THERE SHALL COME THE ROOT OF JESSE: kai palin Esaias legei (3SPAI): estai (3SFMI) e rhiza tou Iessai: (Isa 11:10 Rev 5:5 22:16 Jer 23:5,6)

See the topic Root of Jesse

This quote is from the Messianic promise of the Septuagint (LXX) of [Isa 11:10] when Christ will reign over all nations, Jews and Gentiles.

From the Root comes a Branch thus His name Righteous Branch...

Behold, the days are coming," declares the LORD, "When I shall raise up for David a righteous Branch; And He will reign as king and act wisely And do justice and righteousness in the land. (Jeremiah 23:5)

One would normally use root for the origin (so the root of Jesse would be that from which Jesse derived his being), but the word is not used in this way either in Isaiah or here in Romans. Rather it points to one who springs from Jesse, One for Whom Jesse is the origin.

Although it is somewhat speculative, in Palestine drought would threaten the life of a plant, so that its root system would be its best guarantee of survival. The root came then to be used not only of the root itself but also of that which springs from the root which may explain the picture here in Scripture. In any event, here the One referred to is clearly the Messiah.

AND HE WHO ARISES TO RULE OVER THE GENTILES, IN HIM SHALL THE GENTILES HOPE: kai o anistamenos (PMPMSN) archein (PAN) ethnon, ep auto ethne elpiousin (3PFAI):

NOW MAY THE GOD OF HOPE  FILL YOU WITH ALL JOY AND PEACE : O de theos tes elpidos plerosai (3SAAO) humas pases charas kai eirenes: (Ro 5:2, 3, 4, 5; 8:20, 24, 25; 12:12. 15:4, 12) (Ro 14:17 Jn 14:1,27 Php 4:7 Isa 55:12 Ep 5:18,19)

Spurgeon...
 

Turn back to the fourth verse, and note the expression, "that we through patience and comfort of the Scriptures might have hope; then read in the fifth verse," The God of patience and comfort;" and see how Paul here goes back to that last word in the fourth verse, "Now the God of hope"—
 

Now (de) marks a transition to a new subject, something distinguished in some way from the preceding.

Note also that this sentence is structured as a prayer (O in AAO = optative mood = expresses a wish or in NT a prayer) to the "God of hope", He on Whom both Jews and Gentiles center their hope.

The God of Hope - The truths conveyed are that God is both the origin of hope and the object of our hope ("Who inspires hope and imparts it to His children" Harrison). These great benefits Paul prays for cannot be possessed apart from God, and we can possess them only as He gives them to us. Also note that this joy and peace are intimately related with our continually trusting in Him and His promises (in believing).

Abounding in hope should ideally be the description of every believer. The Christian should be the one in this decaying world who has the inner strength (and Spirit) to look ahead with a contagious enthusiasm. God has filled us with hope, the absolute certainty of future good or that God will do good to us in the future.

JOY (5479) (charas  from chaíro = to rejoice) (cf Ro 14:17-note, Ro 15:32-note) is one of Paul's great themes, with charas being used by him 21x compared to next most frequent use of 9 by John. The Christian life is to be a life of "JOY". It is founded on faith in Jesus, whose life on earth began as "good news of great joy for all people" (Lk 2:10).

Joy is the deep-down sense of well-being that abides in the heart of the person who knows all is well between himself and the Lord and is independent of whether circumstances are favorable or unfavorable (Jn 16:20, 21, 22).

Joy is God’s gift to believers, a fruit of the Spirit (Gal 5:22- note). Nehemiah declared, "The joy of the Lord is your strength" (Neh 8:10). So Paul prays that they would be filled with ALL JOY, that inner gladness and deep seated pleasure. It is a depth of assurance and confidence that ignites a cheerful heart. It is a cheerful heart that leads to cheerful behavior.

Note that Paul prays for this Joy to FILL US [see note on Greek verb (pleroo) under discussion below (Romans 15:14 filled with all knowledge)]. The point is that like wine that fills a man exerts control over him, so too Paul's prayer is that joy and peace would fill the believers in Rome and they would be controlled by this fruit of the Spirit (Eph 5:18-note). The hope we have in Christ is the source of a great part of our joy.

Warren Wiersbe defines joy as — "that inward peace and sufficiency that is not affected by outward circumstances. (A case in point is Paul’s experience recorded in Php 4:1ff-[see notes].) This "holy optimism" keeps him going in spite of difficulties."

Donald Campbell former President of Dallas Theological Seminary says

Joy (chara) is a deep and abiding inner rejoicing which was promised to those who abide in Christ (cf. Jn 15:11). It does not depend on circumstances because it rests in God’s sovereign control of all things (cf. Ro 8:28-note).

Webster's definition reflects the world's view of joy "the emotion evoked by well-being, success, or good fortune or by the prospect of possessing what one desires". Obviously this is not an accurate description of the JOY independent of circumstances that is available to every believer.

Hodge writes

God is called the God of hope, because He is the author of all the well–grounded hope of His people. All hope of which He is not the author, in the heart of men, is false and delusive. The world in general may have hope, but it is false hope. All true hope with respect to the Divine favor is effected in the human heart by God Himself. Not only is God the author of all true hope, but He can create this hope out of the midst of despair. The most desponding are often raised by Him to a good hope through grace; and the most guilty are in a moment relieved, and made to hope in His mercy. How remarkably was this the case with the thief on the cross, and with the three thousand on the day of Pentecost! ...["fill you"] implies that there are degrees of joy and peace in the minds of Christians. Some may have a measure of these graces who do not abound in them. It is a great blessing to be filled with them; and for this blessing the Apostle prays with respect to the Christians at Rome. If there be different degrees of joy and peace, how important is it to look earnestly to God for the fullest communication of these blessings! (Hodge, Charles: Commentary on Romans. Ages Classic Commentaries  or Logos)

PEACE (1515) (eirene from verb eiro = binding or joining together what is broken or divided) (Click word study of eirene) (10 uses of eirene in Romans - Ro 1:7; 2:10; 3:17; 5:1; 8:6; 14:17, 19; 15:13, 33; 16:20) means literally that which has been bound together. It is freedom from disquieting or oppressive thoughts or emotions. Peace in this verse is that inward state of quiet which is independent of circumstances and is that inner attitude which God's Spirit (Gal 5:22, 23-see notes Ga 5:22; 23) gives His people. Note that this peace is only possible when after one has been justified by faith and experienced peace WITH God (Ro 5:1-note). In short peace with God must precede and is the basis for the peace OF God (Php 4:7-note).

Webster defines peace as a state of tranquility or quiet, freedom from disquieting or oppressive thoughts or emotions, harmony in personal relations, a pact or agreement to end hostilities between those who have been at war or in a state of enmity, state of repose in contrast with or following strife or turmoil. (Click for discussion of "gospel of peace" the believer's spiritual "sneakers").

IN (the) BELIEVING: en to pisteuein (PAN):

In believing - Click Romans for 20 uses of the verb pisteuo. Notice that joy is also linked with faith in Paul's letter to the Philippians...

And convinced of this, I know that I shall remain and continue with you all for your progress and joy in the faith (Php 1:25 [note])

Pastor Ray Stedman comments on Paul's phrase in believing :

I want to stress that briefly because I think that we have gone astray in this respect. Oftentimes people come to me, and say, "What is the matter with my Christian life? I have come to a plateau where I seem to be so bored, and nothing interesting is happening, and I have lost all vision and joy and victory in my life. It seems to be so dull and lifeless. What can I do?" For years I think I gave a wrong answer to that. I said to them, "Well, are you reading the Bible?" And usually it turned out that they weren't. Or, "Are you having times of prayer?" And I gave the pat answer which is so easily given by most of us, "What you need is time for prayer and reading the Scriptures -- prayer and the Bible." But I have come to see that this isn't the answer. What they need is to BELIEVE what they read in Scripture, and BELIEVE what they pray--that is the answer. These other things are merely mechanics which make possible the BELIEVING, but BELIEVING is the real answer. It isn't BIBLE READING, or PRAYER, or Christian FELLOWSHIP that unlocks the power of the Holy Spirit. It is BELIEVING what you read or what you pray: When you BELIEVE that Jesus Christ indwells you, when you BELIEVE that he is all that you need, when you BELIEVE that he intends to act through you, then you can act! You discover that all that He is becomes visible through you and accomplishes all that needs to be done. The result is power and joy and peace, as Paul prays here....The God of hope cannot fill us with joy and peace if we don't believe -- which means to act on what we know. But it is when we BELIEVE and act that the power of the Holy Spirit begins to work through us and causes us to abound in hope -- for all around us are the evidences that God is at work accomplishing his purposes in our lives. (Read the full message Power to Please) (bolding added)

In summary, Christian joy and peace are IMpossible apart from trust in Him in Whom they are eminently HIMpossible! (See related topics of In Christ and in Christ Jesus)

THAT YOU MAY (continuously) ABOUND (super abound) IN HOPE: eis to perisseuein (PAN) humas en te elpidi:

Abound (4052) (perisseuo from perissós = abundant in turn from peri = in sense of beyond) means to exceed a fixed number or measure and so to exist in superfluity.

Here we see Paul's purpose - that they would overflowing with hope. The joy and peace given by God should result in an overflow of hope in the believer. 

A common goal in Paul's prayers was that the saints wouldn't be "ain'ts" but that they would ''super abound'' spiritually (Ro 15:13 1Co15:58 1Th 3:12). Paul uses this perisseuo also in Ro 3:7 [note], Ro 5:15 [note]) 

Perisseuo  is used three times in 2Cor 9:8

And God is able to make all grace abound to you, that always having all sufficiency in everything, you may have an abundance for every good deed

In context those who are overflowing in hope by the power of the Holy Spirit have no time to quarrel over nonessentials. Our common hope is a powerful unifying force in the Christian life. For more discussion of the word "hope" see note above on Romans 15:4.

McGee says

This is what a study of Romans should do for you. I trust it has given you joy and peace and that it has strengthened your faith. I trust it has brought hope and power into your life, my friend. This is the benediction that concludes the doctrinal section of the Epistle to the Romans. (McGee, J V: Thru the Bible Commentary:  Thomas Nelson or Logos)

James Witmer writes that...

Paul desired God to fill his readers with all joy and peace (Ro 14:17-note). Joy relates to the delight of anticipation in seeing one’s hopes fulfilled. Peace results from the assurance that God will fulfill those hopes (Ro 5:1-note; Php 4:7-note). These are experienced as believers trust in Him (cf. He 11:1-note). As a result believers overflow with hope by the power of the Holy Spirit (cf. Ro 15:19-note) (Walvoord, J. F., Zuck, R. B., et al: The Bible Knowledge Commentary. 1985. Victor or Logos)

Ray Stedman writes

What a magnificent verse! Whenever I am asked to give an autograph, I almost always include this verse in it. It is such a beautiful expression. Look how much you have got going for you. All the great words of the Christian faith appear here: hope, twice (once it is called "overflowing hope"); and joy, great joy; and peace, calmness and confidence; and trust, belief in a living God; and finally, the power of the Holy Spirit, the invisible force that can open doors and no man shuts them, and can shut and no man opens -- the power of God released among us....What the apostle is urging us to do is to unite on the great positive words of our faith, and that we allow these qualities of hope, and joy, and peace, and trust, and power to be visible when others see us gathered together as Christians. When they hear us talking about each other we are to reflect these qualities, rather than the miniscule divisions and arguments that many of us have. (Read the full message Our Great Example)

BY THE POWER (inherent enablement) OF THE HOLY SPIRIT: en dunamei pneumatos hagiou:

Power (1411) (dunamis) (7x  Romans) refers to inherent power residing in something by virtue of its nature. Here the power Source is the Holy Spirit. His inherent enabling power is the means by which unity will be accomplished. The Holy Spirit supernaturally enables the stronger and weaker brothers to see each other’s positions and to refuse to let their differences mar the unity that they have in Christ. The result is true spiritual freedom and maturity.

What was described as the effect of Scripture in [v4] (have hope) is now attributed to the work of the Holy Spirit. Paul here follows a pattern evident throughout the New Testament in which God’s saving acts are attributed to God’s Word as well as to the work of the Holy Spirit

Cranfield

The existence of this hope in men is no human possibility but the creation of the Spirit of God. (Cranfield, C. E. B Exegetical Commentary on the Epistle to the Romans. Vol 1: Ro 1-8.; Volume 2: Romans 9-16)

Barclay

There are no hopeless situations...there are only men who have grown hopeless about them. (Barclay, W: The Daily Study Bible Series. The Westminster Press or Logos)

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Prepare to Live  (Romans 15:13, 14, 15, 16, 17, 18, 19) - In 1931, Jane Whyte felt she was nearing the end of her life. Her husband Alexander, the famous Scottish preacher, had died 10 years earlier. As she looked at the world around her, she was depressed by the moral and political chaos. There seemed to be no reason for her to go on, nothing for her to do.

At dinner one evening, she sat next to a man who sensed her dejection. "What is your greatest concern?" he asked. "I'm preparing to die," said Mrs. Whyte. "Why not prepare to live?" he suggested.

That was the question Mrs. Whyte needed to hear to break the deadlock in her life. She began to see that God wanted her to live and to touch others for Him. Her attitude changed and within a year she led a Christian outreach team on a mission to Geneva, Switzerland. That trip profoundly affected the lives of many people.

Life can seem overwhelming at times, but God offers us hope. Paul wrote, "May the God of hope fill you with all joy and peace in believing, that you may abound in hope by the power of the Holy Spirit" (Ro 15:13).

Regardless of your age or circumstances, don't despair and "prepare to die." Believers in Christ can prepare to live—filled with hope, joy, and peace. — David C. McCasland (Our Daily Bread, Copyright RBC Ministries, Grand Rapids, MI. Reprinted by permission. All rights reserved)

The hope we have in Jesus Christ
Replaces all despair;
He fills us with His joy and peace
And shows His love and care.—Sper

No one is hopeless who hopes in God.

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Are You Full? (Ecclesiastes 6:7-12, Romans 15:13) - As a boy, I laughed and cried as I read The Adventures Of Tom Sawyer and Huckleberry Finn. I gave little thought to the author of these books, though, until I saw a dramatized version of Mark Twain's life.

Twain had his share of tragedy. He blamed himself for his younger brother's death in a steamboat accident at age 20, and for the death of his only son, who died from diphtheria at 19 months. He grieved bitterly over the deaths of two of his daughters—one from meningitis at age 23 and one from a heart attack at age 29.

But instead of turning to God, Twain became bitter and pessimistic. When he died at 74, he was desperately lonely, unhappy, and hopeless.

Mark Twain had an emptiness that could not be satisfied with money and fame. His success as a writer only increased his misery and sense of loss. His life illustrates the folly of living without God, which is described in Ecclesiastes 6:7, 8, 9, 10, 11, 12. If only he had trusted Christ for salvation and looked to Him for comfort and fulfillment!

Have life's hardships left you feeling empty and bitter, or have they strengthened your relationship with God and made you better? Turn in faith to Christ, and "the God of hope [will] fill you with all joy and peace" (Romans 15:13). — Herbert Vander Lugt (Our Daily Bread, Copyright RBC Ministries, Grand Rapids, MI. Reprinted by permission. All rights reserved)

The sun that hardens clay to brick
Can soften wax to shape and mold;
So too life's trials will harden some,
While others purify as gold. —Sper

Life's trials should make us better—not bitter.

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Happiness And Faith (Romans 8:28-39) - The chorus of the old hymn "At The Cross" concludes with these cheerful words: "And now I am happy all the day!" I don't know about you, but I can't honestly say that just because I know Jesus as my Savior I'm happy all day. I'm a rather optimistic person and I don't let much get me down, but some circumstances don't warm my heart and make me smile.

Troubles may make us wonder: Isn't our faith supposed to make us happy all the time? Shouldn't Jesus shelter us from harm and danger?

Some people teach these things, but the Bible doesn't. God's Word makes it clear that we will have trouble. In Romans 8, for example, the apostle Paul talked frankly about tough times we could face (Ro 8:35, 36, 37, 38, 39). The fact is, Jesus doesn't protect us from all trouble, but His love and His companionship guide us as we go through it.

A more realistic attitude than being "happy all the day" is one stated by a Christian who said, "Now that I'm saved, I'm happier when I am down than I was when I was happy before I was saved."

With Jesus Christ, we can have real joy and make it through even the bad times. — Dave Branon (Our Daily Bread, Copyright RBC Ministries, Grand Rapids, MI. Reprinted by permission. All rights reserved)

The hope we have in Jesus Christ
Brings joy into our heart;
And when we know the love of God,
His peace He will impart. --Sper

Happiness depends on happenings,
but joy depends on Jesus.

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Health-giving Hope (1Peter 1:13-21, Romans 15:13) - It is well-known that our emotions can have a profound effect on our bodies. And the condition of our bodies can affect our emotions.

For example, a 1997 article in the journal published by the American Heart Association points to the negative physical consequences of hopelessness. It essentially said that those who had experienced extreme feelings of despair had a 20-percent greater increase in arteriosclerosis (hardening of the arteries) over a 4-year period. Other studies have also connected hopelessness with heart disease, heart attacks, and death.

The relationship between one's emotional well-being and physical condition, however, is not a modern discovery. In the Old Testament book of Proverbs, we read that "a merry heart does good, like medicine" (Pr 17:22), and that the wisdom found in God's words "are life to those who find them, and health to all their flesh" (Pr 4:22).

A proper relationship to God and His Word can benefit us spiritually, physically, and emotionally. The central concern of the gospel is to bring us into a right relationship with God through faith in Christ. Its blessed byproduct is an abundant life filled with health-promoting hope—the assurance of total forgiveness of sins and eternal life with Christ. — Vernon C. Grounds (Our Daily Bread, Copyright RBC Ministries, Grand Rapids, MI. Reprinted by permission. All rights reserved)

AND CONCERNING YOU MY BRETHREN I MYSELF ALSO AM CONVINCED THAT YOU YOURSELVES ARE FULL OF GOODNESS : Pepeismai (1SRPI) de adelphoi mou: kai autos ego peri humon hoti kai autoi mestoi este (2PPAI) agathosunes: (Ga 5:22 Ep 5:9 2Th 1:11)

Here begins the Epilogue, the personal matters of importance. It is almost as though we have been listening to Paul teaching these Romans, and class is dismissed, and we get a chance to meet the teacher a bit in these last sections.

Paul has just commanded them to accept one another despite their differences and here he is saying in essence "I know you are equipped to carry this out". Paul had somehow become convinced (must have had reports because he had never been to Rome) of the Roman believer's general spiritual maturity (Ro 1:9 [note] "your faith is being proclaimed throughout the whole world.").

Convinced (3982) (peitho)  is perfect tense indicating that Paul had become convinced at some point in time in the past and he remained convinced ("I stand convinced") (cf uses of "peitho" = convinced in [Ro 9:38 {note}, Ro 14:14 {note}]) concerning the church at Rome.

Wuest renders it

I have been completely persuaded with the result that I have arrived at a settled conviction.

Goodness (19) (agathosune [word study] from agathos =benevolent, profitable, benefiting others) describes active goodness, virtue, excellence or beneficence. It is high moral character reflected in to being good in both nature and effectiveness.  Agathosune finds its fullest and highest expression in that which is willingly and sacrificially done for others. It is moral and spiritual excellence manifested in active kindness. Agathosune describes a positive moral quality characterized especially by interest in the welfare of others. Agathosune refers to active goodness as an energetic principle. It is the generosity which springs from the heart that is kind and will always take care to obtain for others that which is useful or beneficial. Thayer says that agathosune is found only in Biblical and ecclesiastical writings.

Wuest writes that agathosune refers...

And so in this passage Paul is speaking of the Roman believers' high moral character and virtuous living. Their hearts were right! As Paul makes clear in [Gal 5:22 [note], cf the other 2 uses Ep 5:9 [note], 2Th 1:11] all virtue is the fruit of the Holy Spirit. But the Spirit can bear fruit only in the lives of believers, such as those in Rome, who are submissive to His divine will and power. They were not perfect, but neither were they spiritually deficient. Their motives were right. They had come to the place where they were motivated by a sense of goodness. The church at Rome was a responsive church, a compassionate church. It reached out to people who were in need. It responded to those who had hurts and burdens and concerns. Whenever a need was shared, there was always a compassionate response.

Paul is saying, in effect:

“In spite of all that I have written to you in this letter—with strong reminders that you were saved solely by God’s grace, made effective by your faith in His Son, with the admonitions for obedience to the Lord, for mortifying the flesh, for holy living, for exercising your spiritual gifts, for serving each other in love and humility, and all the other teachings—I am fully aware of your spiritual maturity and moral virtue, and I commend you for it.”

The only other church he praised so highly was the one in Thessalonica (1Th 1:2, 3, 4, 5, 6, 7, 8, 9, 10 - see notes 1Th 1:2; 1:3; 1:4; 1:5; 1:6; 1:7; 1:8; 1:9; 10).

FILLED WITH ALL KNOWLEDGE: pepleromenoi (RPPMPN) pases (tes) gnoseos:

Filled is perfect tense = they had become filled at some point in the past and have persisted in that state (they "stand filled" or as Wuest has it "having been filled completely full of every knowledge, with the result that you are in an abiding state of fulness"). The believers in this church were doctrinally sound. Not only were their hearts "right" ("full of goodness"), so were their heads! They weren't, as he wrote to one of the other churches, "carried about by every wind of doctrine" that came along (cf, note Ephesians 4:14), but they acted out of an intelligent comprehension of what God wanted them to know. They were not like many Christians today, who seem to spend their whole Christian life just trying to muddle along and do the best they can, hoping that it will all turn out all right in the end. No, these Roman saints didn't do that. They knew something, and they acted from that knowledge. They didn't act apart from it.

Though this is one of the most deeply penetrative theological treatises in the NT, Paul did not write it because these people did not already know the truth that he was giving them! These believers were filled with spiritual insight and perception, knowledge of Christ, God, and the Holy Spirit and of the spiritual need of men.

Filled (4137) (pleroo) (Click for word study of pleroo) (Used 6x in Romans) means to be completely filled (like a glass filled to the brim with water)

Metaphorically pleroo conveys several rich pictures which help us understand the how these believers are filled with knowledge...

(1) PLEROO was often used of wind filling a sail and thereby carrying the ship along. To be filled with the Spirit is to be moved along in our Christian life by God Himself, by the same dynamic by which the writers of Scripture were “moved by the Holy Spirit” (see note 2 Peter 1:21).

(2) PLEROO carries the idea of permeation, and was used of salt’s permeating meat in order to flavor and preserve it. God wants His Holy Spirit to so permeate the lives of His children that everything they think, say, and do will reflect His divine presence. The knowledge of these believers was not to puff them up but to build one another up.

(3) PLEROO conveys the idea of total control. The person who is filled with sorrow (Jn 16:6) is no longer under his own control but is totally under the control of that emotion. In the same way, someone who is filled with fear (Lk 5:26), anger (Lk 6:11), faith (Acts 6:5), or even Satan (Acts 5:3) is no longer under his own control but under the total control of that which dominates him. To be filled in this sense is to be totally dominated and controlled. To be filled with the Spirit (Ephesians 5:18-note) is not to have Him somehow progressively added to our life until we are full of Him. It is to be under His total domination and control. This is in direct contrast to the uncontrolled drunkenness and dissipation in the worship of Dionysius that was alluded to in the first half of (Ephesians 5:18). Thus to be "filled with all knowledge" is to be controlled by that knowledge (What we believe determines how we behave "For as he thinketh in his heart, so is he" Pr 23:7KJV).
 

AND ABLE (being continually enabled) ALSO TO (continually) ADMONISH ONE ANOTHER: dunamenoi (PPPMPN) kai allelous nouthetein (PAN): (Acts 20:31 1Cor 4:14 Col 1:28, 3:16 1Th 5:12,14 2Th 3:15)

Admonish (3560) (noutheteo from nous = mind + tithemi = to place) means literally  to place in the mind. 

Noutheteo conveys the ideas of encouraging, advising and finally warning people of the consequences of their behavior. Admonishing speaks of encouraging counsel in view of sin and coming punishment. In short noutheteo is  giving counsel with a warning involved.

Noutheteo describes an effect on the will and the disposition and it predisposes an opposition which has to be overcome. It seeks to correct the mind, to put right what is wrong, to improve the spiritual attitude.

The "idea is not that of punishment but of a moral appeal that leads to amendment." (TDNT). 

A T Robertson says noutheteo means

"Putting sense into the heads of people. A thankless, but a necessary, task."

Gene Getz writes that
 

Actually, the word noutheteo doesn’t refer to casual communication or normal-type teaching. It implies a definite exhortation, correction, and warning. When this word is used, there is always the implication of a problem. It presupposes an obstacle that must be removed or changed. (bolding added)

Cremer says,

Some degree of opposition has been encountered, and one wishes to subdue or remove it, not by punishment, but by influencing the nous” (mind).

John MacArthur adds that noutheteo
 

is a comprehensive term for counseling. In this context, it refers to coming alongside other Christians for spiritual and moral counseling. Paul is not referring to a special gift of counseling, but of the duty and responsibility that every believer has for encouraging and strengthening other believers.  Tragically, many Christians today have been convinced that competent counseling can only be accomplished by a person who is trained in the principles of secular psychology — despite the fact that the various schools of psychology are, for the most part, at extreme odds with God’s Word and frequently with each other. Although they may profess that “all Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness” (see note 2 Timothy 3:16), many evangelicals — both those who give and those who receive counseling — do not rely on the full sufficiency of God’s Word. (MacArthur, J: Romans 9-16. Chicago: Moody Press or Logos)

Admonishment is a critically important function for the vitality and spiritual health of the body of Christ locally as well as worldwide and a function which is best carried out by those who are full of goodness and filled with knowledge. This is a mark of growing up in Christ, of beginning to take on full stature as a believer, when you are able to help someone else with what you have learned -- "able also to admonish one another."

Warning, admonishing and Biblical counseling are must be carried out (in love) by those in authority in the church. Why is this exhortation so important for the integrity of the local body of Christ?

This same word (noutheteo) is used the the Septuagint (LXX) translation of the Hebrew OT in describing the failure of the priest Eli to rebuke his sons, who "were worthless (Hebrew = belial means worthless, but is a name for the devil. In as sense they were sons of the devil) men; they did not know Jehovah" (1Samuel 2:12). Eli questioned (1Samuel 2:23, 24) but failed to actually admonish his sons. Thus God said

I have told him (Eli) that I am about to judge his house forever for the iniquity which he knew (Eli was not ignorant of their sinful behavior), because his sons brought a curse on themselves and he did not rebuke (LXX = noutheteo) them." (1Samuel 3:13)

Admonishing one another is a vital function which must not be ignored less there be an adverse effect on the spiritual health of the local body.

The NLT although generally a good paraphrase  misses the point on this verse with the translation "able to teach others." Admonishing might involve teaching to some degree but as the English dictionary says to admonish means to

to express warning or disapproval to someone especially in a gentle, earnest, or solicitous manner. To reprove firmly but not harshly. To advise to do or against doing something; warn; caution.

Admonishing is the responsibility of church leaders who should take note of Paul's example. In Acts Paul we read while at Ephesus "Night and day for a period of three years I did not cease to admonish (noutheteo) each one with tears." (Acts 20:31) What was the context of his warning? He was warning them about false teachers, a threat so serious that he even wept because he knew the terrible consequences when false teachers infiltrate. Only by following Paul’s example can the faithful leaders protect Christ’s flock from the savage wolves and diseased sheep who constantly threaten it.

Admonishing is also the responsibility of every believer as clearly indicated by this passage in Romans. In addition we read Paul's second letter the Thessalonians, wherein he exhorts them that

If anyone does not obey our instruction in this letter, take special note of that man and do not associate with him, so that he may be put to shame. And yet do not regard him as an enemy, but admonish (noutheteo) him as a brother” (2Th 3:14,15).

In this verse, Paul provides a much-needed balance to the discipline process, noting that the motive for disciplining sinning believers is love. A parallel thought is found in in Paul's letter tot he Galatians where he addresses the proper attitude of the one who admonishes writing

Brethren, even if anyone is caught in any trespass, you who are spiritual, restore such a one in a spirit of gentleness; each one looking to yourself, so that you too will not be tempted.” (Gal 6:1)

Paul wrote to the believers at Thessalonica to urging them to

admonish (noutheteo) the unruly (in a military context referred to soldier out of rank and behaving in a disorderly, insubordinate manner = anyone who did not perform their duty or follow through on their responsibility), encourage the fainthearted, help the weak, be patient with all men." (1Th 5:14-note) (Red = imperatives or commands - all are present imperatives)

John MacArthur commenting on this verse writes that

For Paul, helping the wayward did not involve some complex methodology or sophisticated psychological counseling program. Instead, other believers were to come alongside them and admonish them. (MacArthur, John: 1 & 2 Thessalonians. Moody Press or Logos)

Writing to the Colossian church (in danger of being taking captive through philosophy and empty deception, according to the tradition of men, according to the elementary principles of the world, rather than according to Christ) Paul wrote

we proclaim (present tense = continually tell with conviction of) Him (Christ in you the hope of glory), admonishing (noutheteo - present tense = continually) every man and teaching every man with all wisdom, that we may present every man complete (full-grown, fully initiated, complete, and perfect) in Christ."  (Col 1:28-note)

Paul uses noutheteo a second time in Colossians exhorting the believers to

Let the word of Christ richly dwell within you (another way of saying "be filled with the Spirit" Eph 5:18-note) with all wisdom teaching and admonishing one another with psalms and hymns and spiritual songs, singing with thankfulness in your hearts to God." (Col 3:16-note)

When God’s Word rules our hearts, His Holy Spirit makes us “rich in the true wisdom” and prepares us to admonish one another, to “teach and help one another along the right road.”  

Admonishing or warning people of the negative consequences of their behavior is the negative side of teaching and it is clear that both of these characteristics are the result of a life overflowing with the Word of Christ.

Certainly, there is in this word some kind of a moral appeal for change in behavior through warning and biblical instruction, or putting sense into the mind with a strong appeal to the will through understanding. In the psychological climate of today where so much stress is placed on professional counselors, these passages (Ro 15:14 [note]; Col 3:16 [note]; 1Th 5:14 [note] teach us an important point. Paul assumes that believers in the body of Christ can and should engage in the ministry of admonishment which is a form of counseling.

J Hampton Keathley III makes an important point...

Two things, however, are required to make us able or competent to warn, instruct, or counsel one another.

(1) We must be full of goodness. Paul teaches us that the believers at Rome were able or competent because they were “full of goodness.” Goodness is the Greek word agathosune which describes that which is good in its character and beneficial in its effect, at least in its goal or purpose. It becomes a synonym for Christ like character in motive, manner, and method. In other words, these believers were those who were mature enough to make sure the plank in their own eye was removed before they sought to remove the speck from their brothers eye (Mt 7:3, 4, 5-see notes Mt 7:3; 7:4; 7:5). They were not out to hurt, but to heal, not to get even, but to help and build up.

Getz writes
 

Christians who are sensitive about their own walk with God are capable—and responsible—to admonish other Christians … It is one of the most difficult exhortations to obey, but it is necessary for the body of Christ to mature and grow.

Admonishment, when done according to biblical guidelines, is not “judging others.” One of the first guidelines was spelled out by Paul: make sure you are “full of goodness” yourself.

Putting it another way, we must make sure we “clean up our own act” before we try to help someone else “clean up” his. Another passage that bears on our ability to admonish one another are the guidelines found in Gal 6:1, 2, 3, 4, 5.

(2) They were also filled with knowledge. Obviously, Paul is talking about an adequate knowledge of Scripture.

Gene Getz writes that...
 

Admonishment must be based upon God’s specific will and ways—not on what we think other Christians should or should not be doing. We must be careful at this point. Many Christians tend to confuse absolutes and non-absolutes. If we exhort Christians in areas that are extra-biblical—areas that are not specifically spelled out in Scripture or specific things that involve cultural standards and practices—then we are in danger of imposing standards contrary to Scripture.

Some churches, groups, or individuals have a list of rigid rules that they often seek to impose on others—the dirty dozen, the nasty nine, the heavenly seven. It varies with the local, but these are invariably doubtful or questionable things, certain practices which a portion of society may condemn as wrong, but they are called questionable because Scripture does not specifically condemn them as wrong. Scripture does set forth principles which guide us in how we should handle such practices. Interestingly, the Apostle Paul preceded this "One Another" injunction in Ro 15:14 with instructions regarding the issue of these questionable practices on which Christians very often disagree." (See J. Hampton Keathley's full article - Positive Commands: The Methods—A Focus on Actions)

Believers in Rome as in every church were expected to help one another to spiritual maturity (each but especially the strong should be an "edifier" Romans 15:2; Romans 14:19). When the body fulfills this role it helps address the pressure upon the pastor who is expected to solve every problem in the congregation, and to counsel everyone first-hand. This was never God's intention. Instead His plan  is that the entire congregation should be involved in the work of admonishing.

Ray Stedman adds that willingness and ability to admonish...

is the mark of maturity, and I think here is one of the great problems of the American church, particularly. Most of us suffer from what someone has described as "prolonged adolescence merging into premature senility." We never grow up -- that is the problem. One of the great admonitions the apostle continually makes is that Christians begin to grow up into Christ so that they are "able to instruct one another." Despite all this, they still needed the ministry of reminding -- they needed the apostle. No matter how much their hearts desired Christ, no matter how much they knew of him, they were still subject to the very human failing of being apt to forget. (See full message The Minister of Jesus Christ) (bolding added)

One another (240) (allelon from állos = another) means just what it says. It is like the sequoia trees of California which tower as high as 300 feet above the ground. You might be surprised to discover that these giant trees have unusually shallow root systems that reach out in all directions to capture the greatest amount of surface moisture. Their intertwining roots also provide support for each other against the storms. That's why the giant Sequoia trees usually grow in clusters. Seldom will you see a redwood standing alone, because high winds would quickly uproot it! That's what "one another" means!