For this kind of issue, you may try the following troubleshooting steps for confirmation:
1. Take the battery out of the laptop & try to turn on the laptop using power adapter only. If the computer turns on, then the battery is defective & needs to be replaced.
2. Make sure your
vaio vgn-cr510e power adapter is not defective. Test the output voltage of the power adapter using a voltmeter. Compare the voltmeter reading to the nominal value written on the adapter or the laptop.
3. Make sure the DC jack is not loose or broken. A loose DC jack will need to be re-soldered. A broken one will need to be replaced.
4. If you have another processor which is compatible with your laptop's model, try swapping processors & test to see if the laptop will work.
5. The power button could be bad
6. The motherboard could be defective & needs to be replaced.
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