You Do Need power plug adapter while traveling from Chile to Cayman Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cayman Islands are A and B and in Chile are C and L. This is how they look like.

Chile

In Chile, outlets of type C and L are in use.

Type C

Type L

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B