Final answer:
Gaius Marius and Sulla gained political power by gaining the support of the common people and bypassing the traditional Roman political system.
Step-by-step explanation:
The 'path to political power' that was common to the careers of Gaius Marius and Sulla was their ability to gain support from the common people and bypass the traditional Roman political system.
Marius rose to power by denouncing the traditional Roman elites and promising to improve the lives of the common people, which gained him widespread support. Once in power, he reformed the army and distributed land to his soldiers, increasing their loyalty to him.
Sulla, on the other hand, used his military power to overthrow his opponents and take control of Rome. He recruited soldiers directly and relied on their loyalty to him rather than the Republic.