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I've recently picked up Russian and in the grammar book I'm using it suggests that for verbs ending in -oвать you remove the ть, the -ова- changes to -у- and then you add the normal endings in the present tense. It also says that for -евать verbs the -ева- is removed in place of -ю- and then you proceed with the normal endings.
However, when conjugating танцевать or ночевать it becomes я танцую or я ночую. I'm not sure why this is the case, or if it's simply an irregularity?
What your grammar book says is mostly correct, e.g. клева́ть (to peck) is conjugated клюю́, клюёшь, клюёт; воева́ть (to be at war) - вою́ю, вою́ешь, вою́ет.
However, when what is left after removing -е́ва ends in ж, ш, щ, ц, or ч, then -у is added instead of -ю so e.g. линчева́ть (to lynch) becomes линчу́ю, линчу́ешь, линчу́ет. This is more of a rule than an exception as it can be seen throughout the Russian grammar. For instance, there is the suffix -ёнок meaning 'baby', 'cub': тигрёнок ('baby tiger'), лисёнок ('baby fox'), etc. After the above consonants it becomes -онок: мышо́нок, галчо́нок. These consonants are called 'unpaired with regard to hardness/softness' (непа́рные по твёрдости-мя́гкости): ч and щ are always soft, while ж, ш and ц are always hard. Thus, the choice of the following vowel (у or ю, о or е) is determined by orthography rather than phonetics. The orthographical tradition prescribes using у (not ю) after these consonants. In fact, it is one of the first rules Russian first-graders learn:
ЧУ-ЩУ пиши́ с бу́квой У.
About тaнцeвать: the original version was танцовать. That has changed but conjugates remain. I checked: Pushkin, Gogol, Lermontov and Turgenev used танцовать in the first half of the 19th century. Also Taras Shevchenko in Гайдамаки: По ночам же танцовать буду с молодцами.
But Tolstoy and Dostoevsky used танцевать in the second half of the century.
The word originated in Western Europe and came to Russia via Poland. In Ukrainian мова it is танцювати. In Polish, it is tańczyć.
You should just remember that it is always чу and цу in all the words except some foreign names or geographical names like Цюрих.
корчевать — корчую,
ночевать — ночую,
гарцевать — гарцую,
танцевать — танцую.
