My question is for example when we read Bismillahir Rahman nir Rahim, since according to the fathah and kasrah sign they are written Bismillahir Rahman nir Rahimi. Ive asked someone before they said the last letter (is read) with fathah/kasrah sound, if you want to read by joining to the next verse.

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Basically, whenever you stop at the end of a word in Arabic, you don’t say the kasra/fatha/dhamna. But if you join the words, then you must say them. So when we say the Bismillah, it’s recited as one sentence without a stop, so after each word in the Bismillah you would have to say the kasra. Yes when you stop at Arrahim, you don’t say the Kasra because you are pausing before saying the following verse