If you're having a problem powering on your camera, first check to see that the battery is fully charged and that you are able to power the camera on using the power button on the inside of the camera. This is done by opening the back door of the camera and pressing the small square button that is between the microSD card slot and the battery. If your camera successfully powers on this way, there may be an issue with the back door of your camera.
The problem with the power button on the back door of your camera can be fixed with a small adjustment. Here is a video that shows you how to make that change. Should you have trouble with this process, contact Support.
If your camera still will not power on, verify that the battery is fully charged. It should take between 4 and 5 hours to fully charge your camera's battery. You may need to try using a different USB port on the computer and/or a different mini-USB cable.