Ciao Bill. In general, we use this shortened version when there are two infinitives. The first one gets shortened. Example: E' difficile poter fare una previsione (It's difficult to be able to make a prediction). It happens with "verbi servili" (dovere, potere, volere). Other verbs can be shortened too, but when we use them in a poem, proverb or simply we want that our speech is more polished: "Esser sinceri oggi non è una cosa positiva" ("Being sincere is not a positive thing nowadays"). It happens with compound verbs too: "Non mi ricordavo di aver visto quel film" ("I didn't remember I had seen that film").

Hello Bill, It is a use and sound issue. They are most used in compound Italian tenses. Actually, such shortened verbs were used in poetry too (even simple tenses), because language sounds more fluent removing the vowel at the end of the verb. But this is a finish. Pronunciating the last vowel is not a mistake. Some examples: -poter fare - aver dato (retta a qualcuno) - andar oltre - guardar oltre (go over appereances) - ''non ti curar di loro'' (this sentence sounds very formal..lol) This increases the register, someway

If you have a noun, you always use "potere". Ex: il potere logora chi non ce l'ha, fare il proprio dovere, i poteri forti ... If you have a verb, shortened form is colloquial or poetical... Ex: dover mangiare e non avere tempo per farlo, dover riposare e non riuscire a prender sonno. "Essere e dover essere" (in this case it's difficult saying "dovere essere") ... Both verbal form "poter" and "potere" are correct.