I can't really add much to what Frenat has already said. He got the right answer first.
We've been operating satellites for decades within the Van Allen belts. Not the same satellites for decades, but we have decades of experience placing spacecraft there and operating them there with predictable and consistent reliability. Yes, if you place a satellite there that was not designed for it, then it won't last as long. But if Kaku conveyed to you the idea that we cannot operate satellites in the Van Allen belts, then he's very wrong.
The Apollo trajectories skirted the Van Allen belts. Dr. Van Allen himself helped to design those trajectories. One of my good friends and colleagues, Bob Brauenig, recreated those orbits numerically from published figures to show their compatibility with trapped-particle models. The links Frenat posted animate a similar demonstration. I prefer mine, accomplished with a donut and some stiff paper. Science should be tastier.
The key is realizing the Apollo translunar orbits were relatively high-inclination orbits, although the inclination is often not depicted in illustrations of it for popular consumption. With a high inclination, you have a three-dimensional solution to the problem of avoiding harsher regions of trapped radiation.
Sadly I've had to clean up after Michio Kaku before. He's a celebrity theoretical physicist, not a practicing engineer, and he harbors a few weird ideas.
