"You cannot loosen a man's tongue with root beer."
Root beer was a non-alcoholic carbonated beverage made from vanilla, molasses, and various root and spice extracts (it could also be made without carbonation). It originated on Earth in the 19th century, and was available across the Federation. On Deep Space 9, root beer was served at Quark's and became a favorite drink of Nog after he was accepted at Starfleet Academy. (DS9: "The Ascent")
Quark and Elim Garak both despised root beer, calling it "vile" and considering it a metaphor for the "insidious" pervasiveness of the Federation; Quark remarked to Garak that root beer was "so bubbly and cloying, and happy... just like the Federation", also noting that "if you drink enough of it, you begin to like it – just like the Federation." Quark asserted that Nog's acquired taste for the beverage heralded the end of Ferengi civilization. (DS9: "Facets", "The Way of the Warrior")
After traveling back in time to 1947, Quark mentioned root beer to General Rex Denning as an example of his knowledge of Human culture. This was a bluff to cover his lack of knowledge. (DS9: "Little Green Men")
However, Quark – never blind to the potential for profit – welcomed the acceptance of Bajor's petition for Federation membership in 2373, predicting sales of root beer alone to increase by five. (DS9: "Rapture")
While Alexander Rozhenko served aboard the IKS Rotarran in 2374, crewmate Ch'Targh mockingly suggested that he might prefer root beer (with a lump of ice cream in it) to Klingon bloodwine. (DS9: "Sons and Daughters")