This is a classic exchange, known by most Christians in the world. Here in the Urantia Book accounting of Jesus' life, we see the "back-story," which highlights clearly the skill with which Jesus was able to dispatch those who would entrap him.
174:2.1 On Monday evening there had been held a council between the Sanhedrin and some fifty additional leaders selected from among the scribes, Pharisees and the Sadducees. It was the consensus of this meeting that it would be dangerous to arrest Jesus in public because of his hold upon the affections of the common people. It was also the opinion of the majority that a determined effort should be made to discredit him in the eyes of the multitude before he should be arrested and brought to trial. Accordingly, several groups of learned men were designated to be on hand the next morning in the temple to undertake to entrap him with difficult questions and otherwise to seek to embarrass him before the people. At last, the Pharisees, Sadducees, and even the Herodians were all united in this effort to discredit Jesus in the eyes of the Passover multitudes.
174:2.2 Tuesday morning, when Jesus arrived in the temple court and began to teach, he had uttered but few words when a group of the younger students from the academies, who had been rehearsed for this purpose, came forward and by their spokesman addressed Jesus: “Master, we know you are a righteous teacher, and we know that you proclaim the ways of truth, and that you serve only God, for you fear no man, and that you are no respecter of persons. We are only students, and we would know the truth about a matter which troubles us; our difficulty is this: Is it lawful for us to give tribute to Caesar? Shall we give or shall we not give?” Jesus, perceiving their hypocrisy and craftiness, said to them: “Why do you thus come to tempt me? Show me the tribute money, and I will answer you.” And when they handed him a denarius, he looked at it and said, “Whose image and superscription does this coin bear?” And when they answered him, “Caesar's,” Jesus said, “Render to Caesar the things that are Caesar’s and render to God the things that are God's.”
174:2.3 When he had thus answered these young scribes and their Herodian accomplices, they withdrew from his presence, and the people, even the Sadducees, enjoyed their discomfiture. Even the youths who had endeavored to entrap him marveled greatly at the unexpected sagacity of the Master’s answer.
174:2.4 The previous day the rulers had sought to trip him before the multitude on matters of ecclesiastical authority, and having failed, they now sought to involve him in a damaging discussion of civil authority. Both Pilate and Herod were in Jerusalem at this time, and Jesus’ enemies conjectured that, if he would dare to advise against the payment of tribute to Caesar, they could go at once before the Roman authorities and charge him with sedition. On the other hand, if he should advise the payment of tribute in so many words, they rightly calculated that such a pronouncement would greatly wound the national pride of his Jewish hearers, thereby alienating the good will and affection of the multitude.
174:2.5 In all this the enemies of Jesus were defeated since it was a well-known ruling of the Sanhedrin, made for the guidance of the Jews dispersed among the gentile nations, that the “right of coinage carried with it the right to levy taxes.” In this manner Jesus avoided their trap. To have answered “No” to their question would have been equivalent to inciting rebellion; to have answered “Yes” would have shocked the deep-rooted nationalist sentiments of that day. The Master did not evade the question; he merely employed the wisdom of making a double reply. Jesus was never evasive, but he was always wise in his dealings with those who sought to harass and destroy him.