What natural resource powered the Institute's broadcasts?

The correct answer is the tidal turbines, which represent a clever use of natural energy derived from ocean tides. In "The Mysterious Benedict Society," the Institute utilized this innovative technology to harness the rhythmic ebb and flow of ocean waters, transforming it into energy. This choice reflects a creative blend of nature and engineering, aligning with the overarching themes in the story of ingenuity and unconventional solutions.

Other energy sources like solar panels, wind turbines, and geothermal energy may be powerful in different contexts, but they are not the specific methods mentioned in the narrative regarding the Institute's operations. The focus on tidal turbines underscores the unique aspects of the Institute that are characteristic of the book's setting and plot, emphasizing forward-thinking approaches to energy and technology.