How do you say 'We search the hotel', as in, for a lost item.
I believe that would be "Wir durchsuchen das Hotel" or possibly "Wir suchen das Hotel durch". "Durch" is "through".
The first one is correct -- Wir durchsuchen das Hotel.
(The verb is accented on the stem, durchSUCHen, so the prefix is not separable.)
Why don't you need a preposition here? Shouldn't it be "Wir suchen nach das Hotel"?
The literal translation of this sentence in English would be "We seek the hotel", which sounds old and weird in English but is normal in German.
suchen includes the prepositon resulting in to search for, to look for.
I read it as "we're searching for the hotel" Duo marked it correct (1st try at sentence).
I agree seek sounds old in english which is why i naturally attempted searching as an alternitive.
I'm not sure why this isn't "Wir suchen den Hotel"... can anybody help?
Why would it be? "Hotel" is a neuter word. "Das Hotel", not "Der Hotel".
My confusion is with the application of the dative case. If "Ich danke der Frau" is dative, I'm not sure why "Wir suchen das Hotel" is not. And I can also now see that even if it were in the dative, it would be "Wir suchen dem Hotel".
Some verbs in German just puts the object in the dative case, like danken, folgen, helfen and so on. There's no clear pattern to it, but thankfully they're rather few so they're easy to memorize. I learned the most common ones very quickly, and it's not a huge issue anymore. When you get a bit further, you'll start to see some verbs taking genitive objects also, which is very unsettling. Most verbs however take accusative objects, like az_p said, so if you're unsure you should always go with that. Even if it's wrong, you'll at least make yourself understood.
That's more to do with danken being a dative verb, which is less common. Verbs usually take an accusative object, so when suchen does in this sentence it's entirely normal.
Read fuzzy889's comment above. In your sentence, you're expecting suchen to take a dative object (dem Hotel is dative case of das Hotel). However, it should take an accusative object (like most verbs). Accusative of das Hotel is still das Hotel.
Suchen can also be combined with nach for much the same meaning. However when you do this, you will need to put the object in dative case because nach is a 'dative preposition'. Wir suchen nach dem Hotel.