How can I boot an Intel Core i7 iMac in 64-bit mode?

I have an iMac running OS X 10.6.8. The processor is a 2.8 GHz Intel Core i7 and the machine has 4 GB of RAM. How can I boot the computer in 64-bit mode (instead of 32-bit mode) so that I can upgrade to OS X 10.9 and run Final Cut Pro 10.3?

To start up with a special kernel check this from Apple: Starting up with the 32-bit or 64-bit kernel in 10.6 and later:

Startup key combination (for current startup only)

If your Mac uses the 32-bit kernel by default, but supports the 64-bit kernel, you can start up using the 64-bit kernel by holding the 6 and 4 keys during startup.

If your Mac uses the 64-bit kernel by default, you can start up with the 32-bit kernel by holding the 3 and 2 keys during startup.

Your Mac will revert to the default kernel the next time you reboot it.

On-disk setting (persistent)

To select the 64-bit kernel for the current startup disk, use the following command in Terminal:

sudo systemsetup -setkernelbootarchitecture x86_64

To select the 32-bit kernel for the current startup disk, use the following command in Terminal:

sudo systemsetup -setkernelbootarchitecture i386

Note: This setting is stored in the /Library/Preferences/SystemConfiguration/com.apple.Boot.plist file and will take effect every time you start up from this disk. If you start up from a different disk, the setting on that disk, or the hardware default, will take effect.

To determine your currently running kernel open Terminal.app and enter:
uname -a
Running with the 64-bit kernel, the output will end with RELEASE_X86_64 x86_64. Running with the 32-bit kernel, the output will end with RELEASE_I386 i386.

In your case you just have to update to Mavericks (Mac OS X 10.9). Mavericks doesn't contain a 32-bit kernel anymore and since your i7-processor is 64-bit, everything just works right out of the box without key strokes or modifying com.apple.Boot.plist.