I just started using Windows 8. One of the first questions that I had was: How do you shutdown a computer in Windows 8?
The answer is:
Swipe or move your cursor from left to right (off screen) on the right part of your screen THEN back onto the display to display a vertical menu. This is a little tricky with a touch pad on a laptop.
Click ‘Settings’ on this menu. Then, on the Settings menu, click on the ‘Power’ icon. This brings up a small menu that has an option to ‘Shutdown’. Click ‘Shutdown’ and your computer will shutdown.
Quite convoluted. Hopefully, this has been improved in Windows 8.1