So I'm struggling a bit with the accusatives and saying "root beer" is confusing me a bit. Root is radiko, and beer is biero. Do I say radiko biero, radikon bieron, or radiko bieron? Or is it something completely different?
I would suggest combining the two into a compound word: either radikbiero or radik-biero (the latter makes the roots clearer). You could also possibly say radika biero, but not radiko biero, since radik' is describing and qualifying biero.
As for accusative forms, adjectives are declined in the same way as the nouns they describe, so the accusative form of radika biero would just be radikan bieron. Only endings change, so the accusative of radik-biero would be radik-bieron.
An international language will always struggle to express purely national concepts - and regional foods and beverages fall strongly into this category. Root Beer is very much an American phenomenon and has not been widely exported like cola and fanta have been. Consequently, one should not expect Esperanto to have a good expression for this - certainly not one that will resonate with anybody not familiar with American culture and the English language.
You'll find suggestions in bilingual dictionaries such as radikbiero or sasafrasaĵo, but use them with caution - and be prepared to explain what they're supposed to mean. (Maybe even include the regional name "root beer.")
I also found these links useful:
Kiel traduki "root beer"? http://www.argilo.net/node/73
Kiel traduki "root beer"? http://argilo.livejournal.com/10203.html?thread=16347
I'd suggest simply checking Vikipedio for concepts like these. It suggests "Radikbiero".
In a compound word like this, the o just takes an -n in the accusative form:
Radikbiero estas bongusta.
Usonanoj trinkas radikbieron.
As others have said, if it were a concept translated with a noun and an adjective, both the noun and the adjective would take the -n, after the a and the o, respectively:
Radikbiero estas senalkohola trinkaĵo.
Mi preferas senalkoholan trinkaĵon.
Root in root beer is not different from solar in solar system: Both are attributes. In Esperanto, when translating root beer, you add the adjectival affix to the Esperanto words for root. So, root beer is translated as radika biero or radikbiero, in the same way solar system is translated as suna sistemo or sunsistemo.
For the accusative, you add the -n affix to the attribute and the noun. So, the accusative of radika biero is radikan bieron, while the accusative of radikbiero is radikbieron.