I understand that the dictionary definition of the expression いけません is "must
not do", and that it can be used in conjunction with say なきゃ in a sentence
such as the following:

試験があるから、勉強しなきゃいけません。 I have to study, because there will be an exam.

いけません seems to 'conjugate' in the short form present tense as follows:

毎日、練習しなきゃいけないんです。 I must practice every day.

And this is where I get a little puzzled. It seems like いけません would be a
conjugation of いける (which presumably would mean something like "to not be
permitted to"), but that doesn't appear to be the case. Is there any logic
behind this, or is it just a peculiarity? Or am I missing something obvious?
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question
grammar parsing
いけません in the short form
> It seems like いけません would be a conjugation of いける (which presumably would
> mean something like "to not be permitted to")

It's true that いけません means something like "can't do", that's why if you want
to say "I must do X" in Japanese you need to actually reword it to "I am not
permitted to **not** do X". Thats why [to say "must" you
use](http://www.guidetojapanese.org/learn/grammar/must):

> Negative te-form + 「は」 (wa) particle + だめ／いけない／ならない  
>  Negative verb +「と」 conditional + だめ／いけない／ならない  
>  Negative verb + 「ば」 conditional + だめ／いけない／ならない

About the combination of なきゃ + いけません, As [this
answer](https://japanese.stackexchange.com/a/42862/36084) clearly explains:

> やらなきゃ is a contracted/collapsed form of やらなければ. いけません and なりません are the
> polite forms of いけない and ならない, respectively.
>
> So やらなきゃいけない (or ～いけません in polite form) is a contracted form of やらなければいけない
> (or ～いけません in polite form). Same goes with the ～ならない/～なりません versions.
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answer