Do ages, generations, administrations, and dispensations come from the same root word?
Yes. The words ages, generations, administrations, and dispensations are said to come from the same root word. In the context of the mystery revealed to Paul, they describe God's hidden purpose not made known in earlier times but now unveiled to His holy apostles and prophets.
Les explains that in this passage, the words ages, generations, administrations, and dispensations all come from the same root word. He notes this is about the revelation of the mystery made known to Paul, which in other ages was not made known. Les connects this to Deuteronomy 29, Luke 18, and John 13, showing God had kept it secret. Now it is revealed to His holy apostles and prophets that the Gentiles are fellow heirs and partakers of His promise in Christ by the Gospel.