Here is a worksheet on the Greek roots arc , archi, archeo, and archae/o. They mean rule, chief, first, and ancient. You can probably see archaeology (i.e. the study of ancient things) growing from the final of the four, but the others are a bit more obscure. There is an element of polysemy in these roots, which may make this worksheet, or at least finding the pattern of meaning in the words on it, a bit more difficult for students.
Still, when you think of words like archenemy, included in this document, archbishop, or archdiocese, suddenly chief and first come into sharper focus. Likewise monarch, archetype and hierarchy, also both present in this worksheet, reinforce those meanings. As far as rule is concerned, many of the political and social positions described by words growing from this root do indeed rule, as well as promulgate rules.
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