This is the 10273rd most frequent Russian word.
большими
"Большими" means "big" or "large" in the instrumental plural form.
The word 'большими' functions as an instrumental plural adjective modifying 'надеждами,' meaning 'great.'
Мы наблюдали закат с большими надеждами на будущее.
We watched the sunset with great hopes for the future.
Here 'большими' describes the plural noun 'мазками' (strokes) and means 'large.'
На выставке были представлены картины большими мазками.
Paintings with large brush strokes were presented at the exhibition.
In this instance, 'большими' describes the plural noun 'улыбками' (smiles), translating to 'big.'