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what part of the computer hardware only retains data while the computer is on
RAM or Volatile storage area is only temporarily memory, requiring electricity to maintain the information stored in it, once the power is off the information is gone.
What part of the computer hardware retains data while the computer is off
A hard disk drive (HDD) is an electro-mechanical data storage device. HDDs are a type of non-volatile storage, retaining stored data even when powered off.
Can I retreive in the computer the data deleted from the Hard disk
Recycle Bin is the first place you should check for deleted data, also you can restore it from a backup or you can try restoring it from a previous version.
Can deleted data be wiped completely in the computer
Even in the worst case that your files are literally deleted and nowhere to be found, the permanently deleted files are actually recoverable with a data recovery program.

Question: Where do I get the recovery program
Self-contained rewrite: Where do I get the data recovery program for recovering permanently deleted files