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A simple sentence has the most basic elements (a subject a verb and a complete thought.
A compound sentence is a sentence that has one or more subject/predicate 
A Complex sentence is a sentence containing one or more dependent clause. 

The difference between them is the way their sentence structure is and what is said in the sentences.
A simple sentence has 1 subject and one verb
A compound has 2 sentences together
A complex can have 2 verbs or 2 subjects
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