Why did the Persians who seemed more powerful lose to Alexander?

Answer 1

Agility and training.

The Persian empire was composed of many different nationalities with different styles of fighting and not always total devotion to the cause of the Great King. For the worst examples charging elephants and chariots with swords sticking out of the wheel hubs are as likely to wipe out your allies as your enemies.
Alexander's Macedonian army was trained to fight in the maneuverable and coordinated Macedonian phalanx formation with the support of his extremely mobile cavalry. He was therefore able to perform stunts like cross right across his front with his mounted friends and sweep around the opposite end of his enemy without the Persians having the ability to responds.