"Я делаю это упражнение."
Translation:I do this exercise.
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Remember that Russian does not have articles (a/the). этот/эта/это/эти are demonstrative adjectives that are used in Russian often to the same effect as using the English definite article "the" to draw specificity to otherwise indistinguishably different objects. As adjectives, they are set in grammatical agreement with the nouns they modify in case, gender, and number (masc, fem, neut, plr respectively).
Many learners of Russian are thrown off by the appearance of это early on as a pronoun in a "this/that is ___" construction. Это is both a demonstrative adjective and a demonstrative pronoun. Remember that the verb "to be" does not exist in the present tense for most declarative sentences. That's why you see Это стол., Это кошка., Это окно., Это дома. Although each of these sentences contain only two words, they are indeed complete sentences. Note that each of them has a different gender and the final one has a plural noun to show the contrast from the adjective это. They are complete sentences because they have a subject and a predicate. Nouns can certainly be in the nominative case in the predicate. It's called a predicate nominative, and it also happens in English. It works the same way as saying Он врач, "he is a doctor". Both nouns are nominative, but the subject of the sentence is "he" and the predicate is "doctor".
You have the right idea that the sentence is specifying a particular exercise by saying "the", but этот/эта/это/эти always equate to this/that/these.
In the English for Russian speakers course they use "этот" and its forms to help the Russian natives learn to use direct articles. I take it that this is mostly a sort of kludgy pedagogical crutch, but I'm not sure if there might be any useful knowledge for learners of Russian (from languages with articles) to be extracted from it.