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I am referring to this PDF or this HTML version of the "verbal roots" in The Concise Dictionary of Akkadian.

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The HTML site says of them pretty much the same thing:

This index of Akkadian roots refers to the principal lemma(ta) in the Concise Dictionary of Akkadian which are verbs in which the root letters may occur. It does not refer to other parts of speech (to which reference is made at the end of each verb). The aleph sign (ʾ) refers to ALL weaknesses, including medial ā, ī, and ū, except for initial w.

I don't get what's going on though. It appears the item on the left is the 3-4 letter "root", while the thing(s) on the right are examples of verbs which use that particular root? Is that it? If so, why are they only showing 1 or 2 examples of verbs using each root, wouldn't there be many more examples in the various texts/corpora? If it's not what I'm saying, what is it? What data is being represented here (left and right).

If the thing on the left is the root, can it be converted into cuneiform script? Or no? Why/why not?

Also, what does the last sentence mean?

The aleph sign (ʾ) refers to ALL weaknesses, including medial ā, ī, and ū, except for initial w.

I am trying to curate this data but I don't quite follow what data it is exactly yet.

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    If you haven't already (and based on this question I assume you most likely haven't), I'd strongly recommend checking out Huehnergard's A Grammar of Akkadian (3rd ed. PDF here on academia.edu). The verbal root is introduced in lesson 3. Sep 3, 2023 at 6:36

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The index contains the root (expressed as the three root letters one after the other) and then the principal lemma(ta) (i.e. entries in the lexicon) in the dictionary.

In Semitic languages, some roots are called "weak". These roots originally contained (and may still contain in some forms) certain "weak" letters which, due to sound changes over time cause the root to take different forms than those expected of regular "strong" roots.

These weak letters typically include the glides y & w, but depending on the specific language may include n (possibly only considered weak in certain positions in the root), alelph (ʾ), and probably others.

A root with its second radical weak is often said to be "hollow". These often behave similarly to strong roots, but with a long vowel between their first and third consonant. These are the weaknesses referred to here as ā, ī, and ū.

Essentially the index is telling you that when listing a weak root, it isn't going to tell you what the weak letter is, and so an aleph could refer to some other weak letter (e.g. a y). The author has decided to mark a root-initial w specifically though, likely because I-w roots (i.e. those with a w as their first radical) have particular behaviour distinct from other I-weak roots in Akkadian.

The Wikipedia page for Arabic verbs gives some illustrative examples of weak roots, their stems, and conjugation. This section here has a table illustrating the major classes of weak root in Arabic (note that Akkadian has a slightly different set of classes of weak root, as it considers more letters weak), and this section here has tables showing the complete conjugation of weak roots of some classes (note that Akkadian conjugation is significantly different from that of Arabic).

Obviously, Arabic grammar is not the same as Akkadian, so you shouldn't expect the forms to carry over exactly, these tables are just to give you an idea of the sorts of changes typical of weak roots in Semitic languages. An Akkadian grammar will be able to go into much more detail regarding the classes of weak roots and their conjugation in Akkadian.

Regarding converting the root into cuneiform, no. It can't be converted directly into cuneiform.

Due to the way Semitic grammar works, the root is an abstract entity consisting solely of consonants. It does not have any intrinsic vowels.

Meanwhile, cuneiform is largely a syllabary (with some logograms and semantic determinatives as well which I will ignore from this point on). That means that it is not possible to write just a consonant, instead one has to write an entire syllable, in most cases consisting of a single vowel and a single consonant (which may either precede or follow the vowel), with some consisting just of a single vowel, and some consisting of a vowel sandwiched between two consonants.

So, if you wanted to write a root out in cuneiform you would have to pick a representative vocalised form of that root. In Akkadian, this would typically this would be the infinitive of the G-stem of the root.

For the root prs "to decide, to cut off", this is parāsum. You then need to work out how this would actually be written in cuneiform (i.e. how to we break the word "parāsum" up into V, CV, VC, or CVC chunks like an Akkadian would) which gets complicated, and would require its own question.

Once you have segmented the vocalised form into V, CV, VC, or CVC chunks, you can convert that into cuneiform using a sign list or an online tool like the one you linked in the question.

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This is meant as an index for students who see an unfamiliar verb in a text, and need to look it up in the dictionary.

Certain consonants in Akkadian are "weak": they assimilate or disappear in particular environments. This means that, when you look at a particular form of a verb, it's not always obvious what the root should be. And if you don't know the root, you can't look it up easily.

So if that happens, you take the consonants you recognize—using ' as a placeholder to indicate "whatever consonant was here has disappeared"—and look up the combination in this table, to find the actual verb root. For example, if I come across a verb that seems to be formed from the pattern _-k-ṣ, this index tells me to look up ekēṣum; if I find something formed from the pattern _-k-š, I should look up akāšum instead.

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