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Installing Ubuntu to an external hard drive has numerous advantages, especially for those dual booters amongst you. Having your Linux system on your external means that you don't have to partition your internal hard drive on your computer, which can save some serious hassle. If you've got multiple computers, this also means you only need the one external to boot Linux on all of them. (Keep in mind that you may need to download more drivers onto Ubuntu to make this work with different computers.)
As external hard drives are becoming smaller in size yet larger in memory capacity, this is becoming an increasingly portable option. You can even have Ubuntu on a flash drive, as opposed to smaller, less fully-featured versions of Linux that were originally designed for the small spaces that were previously all that were available on portable mediums.
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For the latest versions of Ubuntu, you'll need at least 8 GB available on the external hard drive, though the amount required may vary. You'll want to back up any data before continuing on, as the external drive will be completely wiped during the partitioning process. You'll also need a Ubuntu Live CD, which is easy enough to acquire for free if you can't make it for yourself,. Making your own Live CD has its own advantages, as you might prefer an older version of Ubuntu than the ones currently being offered on CD, or some other distro.
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Take the Live CD and boot up your computer with it, according to your bootloader. Once you've got the desktop up and running in Ubuntu, doubleclick the “Install Ubuntu” icon. From there, just follow the prompts. You'll probably want the whole thing dedicated to Ubuntu, though obviously you can partition however you want to suit your particular needs, say, if you want another distro also bootable from the same hard drive or just want some space that isn't dedicated to Ubuntu.
Once this is done, just restart your computer and elect to boot from USB. If all went well, you should be booting Ubuntu!
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Old Computer? No Problem!
Of course, some older computers can't boot from USB, which makes booting Ubuntu from your external hard drive a bit of a problem. This is an issue with the BIOS being out of date which, while not easily updateable, can be done after a lengthy process.
Not interested in updating the BIOS? Get a boot medium. Basically, a boot medium is an alternate boot loader on a CD, floppy, or some other sort of medium that that that version of BIOS supports. You can place GRUB or some other bootloader than does support USB on it. So, essentially, the boot medium is booted by the computer so that you can boot the USB device. This article gives you the details.
Keep in mind that older computers may have trouble running the latest, more CPU-intensive versions of Ubuntu, though proprietary operating systems like Apple or Microsoft of course present even more difficulties. Also, finding the correct drivers for older computers may present a bit of a barrier, though not an impossible one.