Here's why:
* Ancientpower is a move learned by evolution: In Pokemon Emerald, Exeggcute evolves into Exeggutor. Ancientpower is only learned by Exeggutor, not Exeggcute.
* No TM or breeding: Ancientpower isn't available on any TMs in Pokemon Emerald, and Exeggutor cannot be bred to learn it.
Therefore, Exeggcute cannot learn Ancientpower in Pokemon Emerald. You'll need to evolve it into Exeggutor first.