Final answer:
Both the Ottoman Empire and the Mughal Empire consolidated power by establishing strong, centralized governments supported by military dominance and cultural patronage.
Step-by-step explanation:
Both the Ottoman Empire and the Mughal Empire legitimized and consolidated their power by establishing powerful centralized governments. This was achieved through various strategies, including the establishment of efficient administrative systems, the use of gunpowder technology leading to military dominance, and the sponsorship of art and culture. The Janissary system in the Ottoman Empire can be paralleled with the use of slave soldiers in the Mughal Empire, indicating a shared method of military structuring that supported centralized control.
Furthermore, both empires utilized their military and economic strength to encourage artistic and cultural endeavors, which in turn reinforced their legitimacy and consolidated their power.