Final answer:
The Abbasids strengthened their caliphate by promoting the study of mathematics and science, facilitating a cultural and scholarly golden age, and establishing a sophisticated political and administrative system influenced by Persian and Byzantine models.
Step-by-step explanation:
To add to their strength, the Abbasids undertook several strategies. One of these was promoting the study of mathematics and science. They enabled this through the establishment of libraries and translation movements, such as the effort to translate Greek, Persian, and Indian texts into Arabic. This not only preserved previous generations of knowledge but also corrected and expanded upon it, laying the foundations for a golden age of Islamic science and learning. The Abbasid Caliphate also strengthened their political and administrative systems, drawing upon Persian and Byzantine influences, which included employing Persians and their cultural practices, thus enabling a cosmopolitan Islamic society that was fair and just. Their institutions, such as the diwan, and patronage of scholars and translation movements, helped to foster a thriving cultural hub in Baghdad, the heart of their empire.