This group of words will be more or less confusing depending on where you live; in Australia, root and route sound quite different so we get confused more by American TV and movies than in common speech.

Root [noun]: part of a plant that anchors it and sources nutrients for the plant; an anchoring structure or element such as the part of a hair under the skin and the underlying cause of an issue
Carrots and parsnips are both root vegetables.
It is the love of money, rather than money, that is the root of many problems. 

Rout [noun]: an obvious defeat or unruly withdrawal of troops form a battle; a disorganised crowd, often rioting
The General was not sympathetic to his officers after the rout last week.

Route [noun]: a path or means of travel; a designated area or path for a salesperson or franchise; the means of entry (generally used in medical situations)
The cubs carefully planned their route with a compass and map.
Sneezing is a common route of transmission for the flu virus.

To tell root and rout apart, remember that a rout is about the soldiers getting out of a loosing battle.

Of course, I should also mention that there is another (not so family friendly) meaning of the word ‘root’ in Australia – while it may be slang, it is well known and has caused many a laugh when international visitors use the word unwisely!