The Greek word for power here (you are very familiar with it, I think) is dynamis, and sometimes the idea that the gospel is dynamite is appropriate. However, here the focus is not on destroying false religions, clearing a path for God's truth, or helping people overcome persistent habits they cannot shake. Paul explained how power should be understood. The emphasis is not on how it works, but on its effectiveness; it provides something unique—righteousness from God.
The Greek word from which believes is translated is a profound term. Belief in the content of the gospel is only part of its meaning. It signifies trust or personal commitment, conviction to the point of handing oneself over to another person. Although belief involves responding to a truth or a series of truths, this response is not just intellectual agreement but full-hearted engagement with the truth believed. It is not half-hearted.
To believe in Christ means to commit oneself to Him; to trust Christ involves becoming fully engaged with the eternal truths taught by and about Him in Scripture. Such full engagement fosters moral earnestness, dedication, and consecration evident in every part of our lives—24 hours a day, 7 days a week. And there are some aspects of the gospel that a man during Paul's time might be tempted to be ashamed of, especially since Jesus was viewed by the Jews as a criminal, crucified on a tree.