Usually, nothing that bad happens. Starting up should fix most things that were cut short. The exception is when the computer is actively writing or changing data on the hard drive when the power is cut; then the written data will be incomplete. That’s how data can get corrupted. If it’s a word/pdf/general data file that’s corrupted, that file is basically lost. The real bad issue is when this happens to system files that your computer uses to start up; then your PC may have issues start up.
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