A 180 has an HDI-30 SCSI port for use of the PB in Target (here, SCSI) Disk Mode. Mounting the HDD of one Mac on the Desktop of another Mac is a feature common to most of the 68k PBs, later PBs, and most modern desktops and portables. That feature allows you to mount, inspect, and reformat the HDD. However, you need to know that the PB has an HDD (does it audibly spin up?).
You should see the backlight of the display become active, almost immediately after the chime, before hearing the HDD. If that doesn't happen, you may indeed have a problem in the power supply, although a bad (or disconnected) MLB-to-display cable or a bad inverter or inverter cable could also be the cause. The signal ribbon cable from lower case to upper case is quite stiff. If its folding is too tight, it can easily pull out of its mating socket after re-assembly of the case halves.
Does 'I plug the 180 in' mean that you have a working AC adapter? (Leave the battery out of the case until you have sorted out the startup problem.) Is the adapter the correct model for the 180? There are four adapters for the PB1xx series: 15W, 17W, 19W and 24W, and the 180 needs the 19W or 24W. The 180c needs the 24W.
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