Should you remove the battery when you have connected your notebook to power adapter?

You can with most laptops, but then the battery will not be charged. Try it - if the screen goes blank when you pull the battery out then it's off. For most laptops, you only get 1 battery pack out of the box and in order to maximize the battery's life time, once u have a fully charged battery u can remove it when u have an AC power on, so that the battery wont keeps on charging. Cant the laptop detects if the battery is full and cut off the charging? Yes, in most of the cases, but it is still a myth if hooking on both AC and battery together will shorten the battery's life time.