Final answer:
By AD 750, the Arab Empire controlled the Arabian Peninsula, North Africa, and Spain. India was not under their control despite their expansion into the Indian subcontinent up to the Indus River valley. The Umayyad Caliphate led these conquests before the Abbasids took over.
Step-by-step explanation:
The Arab Empire controlled several geographic areas up to the year AD 750. During their swift conquests, the Arabs united the Arabian Peninsula under Islam and expanded their territory vastly. After their victory at Carthage, the Arab armies took over the entirety of North Africa. As their military campaigns continued to make gains, a significant number of African soldiers joined their ranks, some forcibly and others voluntarily, lured by the prospect of wealth from conquests. In 711 AD, the Islamic forces, led by the Umayyad Caliphate, invaded Spain by crossing the Strait of Gibraltar, and soon after, they controlled all major towns in the Iberian Peninsula and made incursions into France.
By 750 AD, at the height of Umayyad rule, the Arab Empire did not extend to India, though they had reached as far as the Indus River valley, which marked the easternmost boundary of their conquests before the year 750. Therefore, while they exercised substantial control in regions such as Spain, North Africa, and the Arabian Peninsula, India was not under Arab Empire control by that time.
The strategic location of Spain and Sicily during the Abbasid period later served as a cultural bridge between the Islamic and Christian worlds, exemplifying a region where people of different religions coexisted. The city of Cordoba in Spain was notably advanced and prosperous.