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There are a few possibilities here,
If the battery is bad, the computer may not work with it installed. Have you tried with and without the battery installed?
Could be a broken or loose power jack. Does the power plug feel loose at the computer or not like it's plugged in all the way?
Could be a bad micro-switch. Does the start button give you 'feedback" ie. Do you feel or hear a click when depressed?
This is only a start of the possibe power issues but seemingly the most common, see if any of this is helpful so far.