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All the dictionaries I've checked say
bu4 zai4 hu5
but should it actually be
bu2 zai4 hu5
I get different answers searching for 不对， some dictionaries say bu4dui4 others bu2dui4, but as far as this phrase goes they all say bu4 zai4 hu, so is this some sort of exception to the tone change rules of 不？
Dictionaries, in general, will not incorporate tone sandhi rules into their pronunciations (of which Mandarin has quite a few)
Wikipedia says the following:
Mandarin Chinese
Mandarin features several sandhi tone rules.
When there are two 3rd tones in a row, the first one becomes 2nd tone, and the second one becomes a half-3rd tone. E.g. 你好 (nǐ + hǎo = ní hǎo)1
The neutral tone is pronounced "low" when following the 1st, 2nd, and 4th tones, and pronounced "high" following the 3rd tone.[dubious ]
不 (bù) is 4th tone except when followed by another 4th tone, when it becomes 2nd tone. E.g. 不對|不对 (bù + duì = bú duì)
一 (Yī) is 1st tone when it represents the ordinal "first," Examples: 第一个 (dìyīgè). It changes when it represents the cardinal number "1" following a pattern of 2nd tone when followed by a 4th tone, and 4th tone when followed by any other tone. Examples: 一个 (yī + gè = yí gè), 一次 (yī + cì = yí cì), 一半 (yī + bàn = yí bàn), 一般 (yī + bān = yì bān), 一毛 (yī + máo = yì máo), 一会儿 (yī + huǐr = yì huǐr).
The rules you need to learn yourself dictionaries will not help you out on this one.
So to answer your question anytime you see 不+4th-tone it will actually be pronounced 2nd tone+4th-tone
The correct one should be bu2 zai4 hu. Unless there is an emphasis for "NOT" CARE, a 4th tone bu4 is then used, but I rarely hear that as a native speaker.
Let's review the tone change rule for 不，
A second tone bu2 is used only when the tone of next character is a 4th tone, i.e. bu2 shi4.
A forth tone bu4 is used if the tone of next character is 1st, 2nd or 3rd tone, i.e. bu4 hao3.
For purposes of "tonality," Chinese doesn't want to have two words with the fourth tone or third tone "back to back." When this happens, the first word in the series takes the second tone instead.
The example of using two words with fourth tones such as 不在 is one where the two words together are pronounced bu2zai4, even though they are pronounced 不bu4 and 在 zai4 when separate.
An example using the third tone is 可 以.
可 is pronounced ke3, and 以 is pronounced yi3 when they are separate, but they are pronounced ke2yi3 when they are together.
