The word exercise has 3 syllables and divided as ex-er-cise. In teaching pronunciation can I divide it as ek-ser-cise since the letter x is the sound of ks together? It seems to make more sense.
That is how the Miriam-Webster dictionary does it:
noun ex·er·cise \ˈek-sər-ˌsīz\
Different reference works have different ways of dividing words into syllables. There's >usually< no issue about how many syllables a word has, but rather concerning where the breaks fall. My own practice is to honor the etymology where this is possible, as in lob-o-to-my, not lo-bot-o-my [Merriam Webster], or psy-cho-log-ist, not psy-chol-o-gist [American Heritage], Is "bot" a proper word element? Is "chol"?