What is the root of the word:
I think it is "Converse" but I am not sure.
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It depends on how you define the concept of "root". Are you talking about something that is supposed to be part of the synchronic morphophonological system of present-day English, or are you talking about the diachronic derivation of the word? Sometimes its hard to separate these two concepts, but in many theories of morphology, they are theoretically distinct.
I think almost any present-day English speaker would realize that the noun conversation is related to the verb converse. So we probably can say that either converse is the root, or the root is contained in converse.
From an etymological perspective, the "con" in this word is a prefix (the Latin prefix con- roughly meant "with"; it is related to the Latin preposition cum "with").
There are some arguments for analyzing con- as a prefix not only in Latin but also in the morphophonological system of present-day English, but this issue is somewhat debatable. It's not necessarily obvious to all English speakers that con- can be considered a prefix in the verb converse.
Even though converse seems like it cannot be divided into a prefix con- and a free morpheme verse (verse does exist as an independent word in English, but not as a verb with the correct semantics to serve as a base for the verb converse) we could consider -verse in this context to be a bound morpheme of present-day English: there is another verb ending in -verse that can be analyzed as starting with a prefix, re-verse, and there are a number of words in other categories that end in -verse such as the adjectives di-verse, per-verse, in-verse, ad-verse. The semantic relationship between these words is not really obvious to a speaker of present-day English, but you could argue that they are morphologically related nonetheless.
Many speakers reduce the vowel in the first syllable of the verb converse, pronouncing it something like [kənˈvɜrs]. But we generally don't hear vowel reduction like this in word-initial unstressed closed syllables that did not originally come from Latin prefixes: consider the typically unreduced pronunciations of the first syllables of words like monsoon, pontoon, trombone, Montana, pontificate, monstrosity, spontaneous. This special phonological behavior of words that start with a com- or con- derived from the Latin prefix is mentioned in Greg Lee's answer to the following question: The pronunciation of words which begins 'con' and 'com'. So it seems con- may have a special status in the phonology of present-day English, which might be explained in terms of it being a prefix. But one argument against analyzing con- as a prefix in the synchronic morphology of present-day English is that it has almost no productivity (unlike, say, re-). The most recent word that I could find in the OED that was created by prefixing con- to something else is the obscure scientific term conalbumin, for which the first OED quotation is from 1900.
So the root could be considered to be verse (or -vers-) if you consider con- to be a prefix. Another term that I have seen that seems like it might be applied to vers is "paleo-morpheme", which is defined by Marina Yaguello in Language Through the Looking Glass: Exploring Language and Linguistics as a word-part "whose status and meaning are impenetrable for many speakers" (p. 39 of the edition adapated from the original French by Yaguello and Trevor A. Le V. Harris that was published in 1998). Another resource I found online uses the term "pseudo-morpheme" or "quasi-morpheme" to refer to things like the con- and -ceive in conceive.
If you get even more abstract, you could consider vers to be an "allomorph" or even a derived form of the root morpheme vert, which was the more basic form in Latin (in Latin, when a root ending in t was followed by a suffix starting in t, the two ts were replaced with ss or s). However, I think it's quite questionable whether things like this t/s alternation in words taken from Latin really constitute part of the synchronic morphophonological system of present-day English.