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How do you say beef in Spanish? Is is "carne de vaca" or a more common way to say it?

"carne de res" is more common, though you can get away with just "carne". In most spanish speaking countries, default "carne" (which is simply "meat", though there are many kinds of meat) will almost always mean meat from a cow since coincidentally, any other meat is not as common; perhaps even due to geographical considerations (I cannot think of a spanish speaking language where it'd be more common to eat, say, goat meat, than beef).

Se dice carne y puede ser de diferentes tipos de animales.

carne de res
carne de puerco
carne de chivo
carne de caballo etc..................

is just meat

In Spain is more common to say "carne de ternera". :)

We usually say "carne" for beef, "pollo" for chicken, "cerdo" for pork, "cordero" for lamb
and if you want to be spesific just say "carne roja" or "carne de vaca"