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Excuse is pronounced as /ɪkˈskjuːz/ while sixteen is pronounced as /sɪksˈtiːn/. You see a ' in the middle of k and s in excuse while ' is after both k and s in sixteen. Why?

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English generally follows the Maximum Onset Principle, which says that syllables are broken so that a consonant is placed into the following syllable as long as it is phonologically valid. So for example, in excuse, you cannot place the /k/ in the second syllable because a word (and thus a syllable) cannot start with ksk in English, thus /ɪkˈskjuːz/.

If there is a morpheme boundary in the word, syllables are broken at the morpheme boundary rather than according to the maximum onset principle. This happens with sixteen, where the word is broken /sɪksˈtiːn/ rather than /sɪkˈstiːn/, because 'six' and 'teen' are morphemes.

People may not always agree about syllable breaks. For example, in my speech I break waitress into syllables as /ˈweɪt.rɛs/, I assume because I perceive 'wait' as a morpheme, but according to dictionaries it is broken /ˈweɪ.trɛs/ (see the discussion in the comments of the linked question).

You could ask why 'ex' doesn't count as a morpheme in excuse. I certainly don't know the answer to that, and I'm not sure whether anybody else does, either. But I do pronounce it with a syllable break after the /k/. Whereas ex-king, for example, would be pronounced with the break after ex.

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The sign / ' / is the primary stress indicator : it is placed in front of the main stressed syllable of the word.

In excuse the stressed syllable is /skjuːz/.
In sixteen - in the pronunciation you give - the stressed syllable is the last one /tiːn/.

Sixteen is often pronounced with a secondary stress syllable, shown by the sign /ˌ/ −> /ˌsɪksˈtiːn/ or even with two stressed syllables /ˈsɪksˈtiːn/, "according to sentence stress" (quoting phonetician Daniel Jones).

You will find the explanation of IPA sounds and symbols on Wikipedia.

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Then why it's /ɪkˈskjuːz/ not /ɪks'kjuːz/? –  Phương Nguyễn Dec 26 '11 at 12:54
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Both (/ɪkˈskjuːz/ and /ɪksˈkjuz/ ) are found on wiktionary. I checked in three different pronouncing dictionaries: D. Jones (1967) gives /ɪksˈkjuːz/, Lewis (1972) and J.C Wells (1990) give /ɪkˈskjuːz/. Although IPA was not in use in the 18th century John Walker would probably have used Jones's transcription, see p. 63, §477. I agree with you /ɪksˈkjuːz/ seems more logical, I wonder if that has to do with the fact we're both non native speakers. –  Laure Dec 26 '11 at 14:35
    
And the American Heritage Dictionary breaks sixteen after the /k/, so if they used IPA it would be /sɪkˈstiːn/. –  Peter Shor Dec 26 '11 at 15:04
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Both /ɪkˈskjuːz/ and /ɪks'kjuːz/ are allowable, since /kj/ and /skj/ are valid syllable onsets, and /k/ and /ks/ are valid syllable codas. Either one will work, and people vary in their pronunciation and perception of them. On the other hand, the vowel in /ɪk/ in excuse is unstressed, and therefore is often (by many speakers, always) reduced to a central vowel [ɨk]; Americans, especially, centralize most unstressed vowels to [ɨ] or [ə] –  John Lawler Dec 26 '11 at 15:33
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@John Lawler: except sometimes in "excuse me", pronounced sarcastically. –  Peter Shor Mar 1 at 23:15
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