Many words are spelled with -ise in British English and -ize in American English:
But exercise can only be spelled with -ise, never with -ize.
Is there a good reason for this?
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Those are not the same kind of word. Your mistake is thinking that English spelling has something to do with its pronunciation, and this is in error. It’s all about the etymology. Consider also:
abscise, advertise, advise, affranchise, apprise, arise, chastise, circumcise, comprise, compromise, despise, devise, enterprise, exercise, expertise, franchise, frise, galliardise, incise, merchandise, revise, sunrise, supercise, supervise, televise.
And contrast that set with this set:
analyse, autolyse, breathalyse, catalyse, chemolyse, dialyse, glycolyse, haemolyse, heliochryse, histolyse, hydrolyse, lyse, metanalyse, paralyse, photocatalyse, photolyse, plasmolyse, proteolyse, pryse, psychoanalyse.
It’s the etymology that counts, not the pronunciation. It turns out that the -ize ending of words comes, per the OED, from:
Cognate with French -ise-r, Italian -izare, Spanish -izar, < late Latin -izāre, -īzāre, < Greek -ίζειν, formative of verbs.
Whereas exercise has a wholly different provenance:
Middle English exercise, < Old French exercice = Provençal exercici, exercisi < Latin exercitium, < exercēre to keep at work, busy, employ, practise, train (compare exercise v.), < ex- (see ex- prefix1) + arcēre to shut up, restrain.
The note at -ize is quite long:
The Greek verbs were partly intrans., as βαρβαρίζειν to play the barbarian, act or speak as a barbarian, side with the barbarians, τυραννίζειν to side with the tyrants, partly transitive as καθαρίζειν to purify, clean, θήσαυρίζειν to treasure up. Those formed on national, sectarian, or personal names were primarily intransitive, as Ἀττικίζειν to Atticize in manners, to speak Attic, Φιλιππίζειν to act or speak for Philip, to philippize, Ἑλληνίζειν to ‘do’ the Greek, act as a Greek, speak Greek, Hellenize; also, to make Greek. A few words of this form connected with or used in early Christianity, were latinized already in the 3rd or 4th cent. by Christian writers: such were βαπτίζειν baptizāre, εὐαγγελίζειν euangelizāre, κατηχίζειν catechizāre, σκανδαλίζειν scandalizāre, ἀναθηματίζειν anathēmatizāre, χριστιανίζειν christiānizāre, ἰουδαίζειν iūdaizāre. Others continued to be formed both in ecclesiastical and philosophical use, e.g. canōnizāre, dæmonizāre, syllogizāre (Boethius Aristot. Anal.); and this became established as the normal form for the latinizing of Greek verbs, or the formation of verbs upon Greek analogies. In medieval Latin and the modern languages these have been formed also on Latin or modern national names, and the use has been extended to the formation of verbs from Latin adjectives or ns. This practice probably began first in French; in modern French the suffix has become -iser, alike in words from Greek, as baptiser, évangéliser, organiser, and those formed after them from Latin, as civiliser, cicatriser, humaniser.
Hence, some have used the spelling -ise in English, as in French, for all these words, and some prefer -ise in words formed in French or English from Latin elements, retaining -ize for those formed < Greek elements.
But the suffix itself, whatever the element to which it is added, is in its origin the Greek -ιζειν, Latin -izāre; and, as the pronunciation is also with z, there is no reason why in English the special French spelling should be followed, in opposition to that which is at once etymological and phonetic. In this Dictionary the termination is uniformly written -ize. (In the Greek -ιζ-, the i was short, so originally in Latin, but the double consonant z (= dz, ts) made the syllable long; when the z became a simple consonant, /-idz/ became īz, whence English /-aɪz/.)
This is to be contrasted with the -ise suffix from exercise, of which the OED says:
Suffix of ns., repr. Old French -ise, properly:—Latin -ītia, but also, in words of learned formation, put for Latin -icia, -itia, -icium, -itium, as in Latin justitia, judicium, servitium, Old French justise, juise, servise. Hence it became a living suffix, forming abstract nouns of quality, state, or function, as in couard-ise, friand-ise, gaillard-ise, marchand-ise. In the words from Latin, -ise was subsequently changed in French to -ice, as in justice, service, in which form the suffix mostly appears in English, as in justice, service, cowardice; but -ise is found in franchise, merchandise, the obsolete or archaic niggardise, quaintise, riotise, truandise, valiantise, warrantise, and in such barely-naturalized words as galliardise, gourmandise, paliardise; also, in exercise, French exercice, Latin exercitium. Native formations on the same type are inconvenientise, sluggardise.
So you see, the spelling is all about the etymology, not about the pronunciation.