Final answer:
The mode where RAM stays powered while other systems are off is called sleep mode or suspend to RAM, allowing quick resumption of activity.
Step-by-step explanation:
The mode in which RAM (Random Access Memory) stays powered up while all other systems are shut off is known as sleep mode or suspend to RAM. When a computer enters sleep mode, the system saves the contents of the RAM and turns off most of the computer's components, dramatically reducing power consumption. However, because the RAM remains powered, the system can quickly resume where it left off when the user wants to start working again.