Question

Which is the independent clause in the sentence? The treasures that archaeologists found gave clues that King Tut's tomb might exist in the Valley of the Kings in Egypt.

Answer 1

Actually, the independent clause in this sentence is 'The treasures that archaeologists found gave clues'. The other sentence is the subordinate, because it further explains what the clues are. You cannot start the sentence the way this one is written 'that king tut's tomb might exist in the valley of the kings in egypt' because it is incomplete. 

Answer 2

The independent clause in the sentence is the clause that can work alone as a simple sentence. In this case it is "King Tut's tomb might exist in the Valley of the Kings of Egypt." The first half of the sentence, "The treasures that archeologists found gave clues that", does help the sentence but the latter half makes sense on its own.