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Is the underlined word acting as an adverb or noun?

I like to sing inside the shower.

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Neither. It's a preposition.

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Rover_KE is of course correct that "inside" is a preposition.

You may also be interested to know that the preposition phrase "in the shower" serves as an adjunct (sometimes called an 'adverbial'), where it modifies "sing".

Students: Are you brave enough to let our tutors analyse your pronunciation?
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I always hear and read eg I like to sing in inside the shower'.

Students: We have free audio pronunciation exercises.