Short Answer 
Machiavelli argues that to secure a conquered republic, a prince must ensure its complete destruction or maintain direct control to prevent rebellion, which often arises from ideals of freedom and tradition. He distinguishes between principalities and republics, noting that republics require more stringent measures to suppress resistance due to their ingrained spirit of self-governance.
Step 1: Understanding Rebellion in Conquered Republics
Machiavelli emphasizes that the key to securing a previously free state is to ensure its complete destruction. This is critical because if a prince fails to take decisive action, he risks facing rebellion and potential ruin. The roots of rebellion often lie in the ideals of freedom and tradition, which continually threaten the stability of the conquered state. Therefore, it is essential for a prince to recognize these factors and act accordingly.
Step 2: Comparing Principalities and Republics
In his analysis, Machiavelli differentiates between principalities and republics. He notes that principalities, where citizens have always been ruled by a prince, are generally easier to control after the previous ruler has been overthrown. This is because the populace is accustomed to following a ruler and lacks experience in self-governance. Conversely, republics foster a spirit of resistance, making it crucial for a new ruler to take strong actions to suppress dissent.
Step 3: Strategies for Managing Conquered States
To maintain power over a conquered republic, Machiavelli advises two main strategies: complete destruction of the republic or direct occupation by the prince. This dual approach serves to eliminate entrenched sentiments of hatred and revenge within the populace. By either dismantling the republic entirely or having a strong presence in the region, a prince can significantly reduce the risk of insurrection and establish a more stable rule.