This is the famous battle cry "Axes of the Dwarves!" Tolkien states in Appendix F that the Dwarves had used it "on many a field since the world was young." Gimli used it at the Battle of Helm's Deep as well, along with the other half Khazâd ai-mênu!
We know from several places that Khazâd is the Dwarves' name for their own race. This would then be a proper noun. Arabic treats proper nouns as syntactically and semantically definite (as in the Dwarves). This is even when the word doesn't have a definite article in front of it. We see no such thing here, even though the translation is "the Dwarves". I would suggest that the word "the" in the translation is simply due to English. If we were to say "Axes of Dwarves!", the question would be "Which Dwarves?" Instead, in English when we talk about an entire, specific group like this, we add "the". If Khuzdul follows Arabic's example and treats proper nouns as definite, regardless of the presence of a definite article, then indeed Khuzdul could get away with not showing the definite article here and still having the English translation be "the Dwarves". I considered that the definite state may have been indicated by vowel changes, such that we would have khazad "Dwarves" and Khazâd "the Dwarves". However, Tolkien says that khazad is actually the "composition form", and this fits better with what we know of Hebrew and Arabic. Khazâd is probably used for both the definite and indefinite, and which state will be determined by context from the speaker. The definite article, assuming it exists in Khuzdul, might actually be used when speaking of a specific subset of Dwarves, as in "the Dwarves who ransacked Menegroth".
In Parma Eldalamberon XVII, pg 35, we find "the dwarf-name for themselves was KhZD with various vocalizations: apparently sg. Khuzd-, pl. Khazâd, form in composition khăzăd." (Here, the ă denotes a short vowel.) We know then that Khazâd is plural, and that here is is not in composition form. Because it is the second element in the phrase, which is most probably an example of a Semitic-style construct phrase, it makes sense that it would not appear in composition form (see more below on this point). We also see the exact same form in the companion phrase Khazâd ai-mênu, which shows that in both places it is in nominative form.
Khazâd is then analyzed as "(the) Dwarves" and plural, nominative, and definite (as a proper noun) or indefinite (depending on context).
The first word, baruk, is translated by Tolkien at one point simple as "axes", without any additional information. In Parma Eldalamberon XVII, pg 85, he says "...baruk being the plural of bark 'axe', ...". We can see that it's plural, but again there's no hint of case or state. Given that the translation of the phrase is "Axes of the Dwarves!", this looks to be similar to the construct state in Semitic languages. In Biblical Hebrew, in particular, the first word of a pairing is placed in the construct state (or "bound form") in order to indicate a genitival relationship, as in "X of Y". It's conceivable that the pattern CaCuC is a construct or genitive form of CaCâC from Khazâd. However, the singular bark has the vocalization CaCC, different from CuCC of Khuzd. That, and because Tolkien doesn't provide additional information, indicates to me that baruk is a different template from Khuzd / Khazâd.
It's more likely that Khuzdul follows the Hebrew formation of the construct state, which is to reduce the vowels. That vowel reduction is due to a loss of stress because the two words almost become like a compound word. Indeed, the two elements of a construct pair in Semitic languages cannot be separated by any other word, adjectives included. Based on this line of thinking, it's very possible that the indefinite is barûk, similar to shathûr from Bundushathûr. Ssince Tolkien doesn't say anything about baruk being the composition form, I think it's most likely baruk is the normal, indefinite state for "axes". Based on this form and looking to Hebrew's rules for vowel reduction, the composition form would also be baruk. With the two states being identical in form, it makes it more likely that Tolkien would not have offered further clarification on the "axes" translation. As in (Biblical) Hebrew, the construct phrase may, in some instances, be indicated merely by apposition of nouns (placed side by side) with no further apparent changes. The stress in pronunciation may still change slightly during speech.
As a side note, shathûr "clouds" and the vocalization pattern CaCûC may not be a true plural anyway. See Bundushathûr for more information.
I will say baruk is "axes" and is plural, nominative, composition, and also has the same form as when plural, nominative, indefinite.
The whole phrase "Baruk Khazâd!" can thus be interpreted as a construct phrase similar to that found in Biblical Hebrew, used to indicate genitive relations, as in "X of Y", such as ownership, as is the case here.
The Lord of the Rings, The Return of the King, Appendix F
The War of the Ring, pg 20
Parma Eldalamberon XVII: Words, Phrases and Passages, pg 85