Organic Chemistry Nomenclature !?

The "a" is added if the "t" of pent would be followed by another consonant. It is simply added to make it 'sound better'.
Similarly the vowel can be omitted if there will be another vowel immediately after. So it's not really a matter of chemistry, just a matter of euphony.

So, ignoring the numbers, it would be "pentenol", but "pentadienol"
Similarly, it is "ethanoic acid", but "ethanedioic acid"; "propanol", but "propanediol".

According to IUPAC Nomenclature of organic compuds :
The authorized "root-word" in chemistry is "pent", not "penta" and "-ene", not "-en".
But, in case of alcohol, "pentan", "hexan", "propan", etc. is used. For example pentan-2-ol, propan-1-ol, etc.

Both are wrong.
Pentan-2,4-dien-1-ol is correct.

That is simply a typo, in an alcohol it should always be penta

Edit. TheHex is correct about it being pentan is most scenarios. However, Pentan would not come at the beginning in this scenario, as you have the 'dien' which provides the n. The correct answer is still Penta-2,4-dien-1-ol, not Pentan-2,4-dien-1-ol.

it somewhat is only a typo, in an alcohol it might constantly be penta Edit. TheHex is right approximately it being pentan is maximum circumstances. in spite of the undeniable fact that, Pentan does no longer come before everything in this difficulty, as you have the 'dien' which delivers the n. the surprising answer remains Penta-2,4-dien-a million-ol, no longer Pentan-2,4-dien-a million-ol.