You Do Need power plug adapter while traveling from Cocos (Keeling) Islands to Chile.

Here is Why You Need An Adapter

The outlet and plug type used in Chile are C and L and in Cocos (Keeling) Islands is I. This is how they look like.

In Cocos (Keeling) Islands, outlets of type I are in use.

Type I

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

Type C

Type L

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