This example shows how to organize FastAPI routes using routers. First, you import FastAPI and APIRouter. Then, create a router instance and add routes to it, like a GET route at '/hello'. Next, create the main FastAPI app instance. Include the router in the main app using app.include_router(router, prefix='/api'). This means the router's routes are accessed under the '/api' path, so '/api/hello' works but '/hello' does not. When the app runs, requests to '/api/hello' return the expected JSON message. Requests to paths not defined return 404 errors. Using routers helps keep your app organized by grouping related routes and optionally adding prefixes.