Why it turn off even though I already turn it on many times

@idk70374 Start off with the easy things. Disconnect the power from the computer by unplugging the power cord from your computer. Disconnect all external devices from the computer. Remove the battery Now hold the power button depressed for about 30sec-1min. Reapply the AC power adapter and see if your computer does show any sign of life.

Next you want to check your power supply. Make sure it has the proper voltages. Check to make sure that connectors are not bend etc. Check the DC-in jack of your laptop. Make sure there is no corrosion, debris etc. that prevents it from working. Clean it of any debris etc, use a few puffs of air to clean it out. If none of this works, get an ammeter and see if your laptop draws any amperage from the charger. After that, you need to check and see if your motherboard even gets power. For that of course, you need to open it up. Post Some good pictures of what you see. That way we can see what you see. Adding images to an existing question

Of course, you should also consider replacing the battery. However, it bad battery should not prevent your computer from coming on, if it is connected to the AC adapter. Looks like yours is connected to the AC adapter.

It is also possible that this is caused by a bad RAM module. Remove the RAM, clean the contacts with a pencil eraser. Do it gentle so as to not destroy the module. Reinstall it and restart. If you have more than one module, remove it and just leave one connected.

Here is the Compaq Compaq CQ40 Maintenance And Service Guide That will help you to work on your laptop should further steps like motherboard repair :-) etc. be necessary. Let us know what you find out.