Casio CDP-100 is not getting power. I believe that it is a bad A/C adapter. What is the problem?

Troubleshooting on a similar model here, but unless you are an electronics engineer, you will need a Casio Service Centre

https://www.ifixit.com/Wiki/CASIO_CDP-230R_Troubleshooting

https://www.google.com/search?q=Casio+COP-100+Digital+piano+no+sound

..

What kind/type of your Casio 100?
Casio MT-100,LK-100,CA-100,PT-100,CDP-100 ?

Assuming that the adapters are 12v negative(-) center and an incorrect positive(+) center was never used. That would cause a shorted diode in the power supply. The next most common problem would be a cracked trace or solder joint inside the keyboard at the power jack. Other problems require more detailed troubleshooting. A wire soldered across the cracked trace(if that is the cause of the trouble) is relatively easy if you own a soldering iron. If not bring the keyboard to a technician.

I only have the one adapter.

Casio CDP has Auto Power Off: 30 minutes after last key operation, Auto Power Off can be disabled.
Please read the manual.

If it turns itself off while playing, I would try to buy a new AC Adapter to test. DC cable of AC adapters are often broken internally after much "bending".

usual problem with casio
the solder broken in dc jack in to the piano

Test the adapter with a meter... Most probable cause is a bad power connector. Often the solder at the input power connector breaks loose is only problem. If you are not electronically adept, best to take it to a shop.

First thing is to get a meter and verify that the power adapter is producing voltage that it is supposed to. You can get inexpensive meters at Harbor Freight for example. I am not sure what you mean by "several outlets"... I HOPE that is not several on the Privia because there is ONLY ONE correct place to put the adapter cord into the Privia.
If you mean wall outlets, verify them by plugging in a known good lamp or appliance to check the receptacle is good.

Try this link for the manual:http://support.casio.com/pdf/008/CDP100_e.pdf