I have a phrase and I'm concerned with grammar. Which one would be more proper?
et ego non curae
et ego non curo
Phrase meaning would be "I don't care."
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The standard way of saying "I care for X" in Latin is:
X (nominative) mihi est curae.
De X (ablative) mihi est curae.
This is a so-called double dative construction, and literally means "X is a concern to me" or "It is a concern to me about X." The proper way of saying "I don't care [about that]" is thus:
Mihi non est curae [de hoc].
Here are some other options, mostly from my favorite Latin playwright. My translations are literal, but you could easily use some version of "don't care" instead:
Ista non curo. (Cicero, Epistulae ad Atticum 184.108.40.206)
I don't care about these things.
Istuc quidem edepol nihil est. (Plautus, Miles Gloriosus 1.1)
That is absolutely nothing.
Tu [hoc] hau magni péndis. (Plautus, Pseudolus 1.1)
You really don't think that is worth much.
Quod dixi flocci existumat. (Plautus, Mostellaria 1.1)
He doesn't give a straw for what I said.
Depending on the context, the following can also be a way of "not caring":
Nugas blatis / garris (Plautus, Curculio 3.1, 5.1)
You're spewing nonsense.
Quin tu is in malam crucem? (Plautus, Curculio 5.2)
Why don't you go to hell? (lit. "a bad cross")
A great little phrase used was flocci non facio, rendered by the Victorians as "I care not a straw for" but really means something closer to "I don't give a rat's ass." It was used by both Cicero (Att. 13.50.3) as well as Plautus (all over the place) and Terence (Eun. 3.1.21), so it's clearly a common and idiomatic phrase.
The main difference in your phrases is that "curae" is a noun and "curo" is a verb. Therefore, et ego non curae is not exactly a complete phrase because there is no verb.
et ego non curo is correct and means "I don't care", but you could also say et ego non curam do, which means "I don't give a care". It just depends whether you want to use a verb or noun to translate care.
Additionally, you don't directly have to include ego because that is indicated by the verb ending.