The sentence has two clauses: "Please watch the babies" and "while Mother takes a nap". (You didn't underline a clause so I'm going to explain both of them).
First Clause: "Please watch the babies" This is an independent clause. You can identify independent clauses when you read it and it has a full idea already. It can stand without another clause to back it up.
Second Clause: "while Mother takes a nap" This is a subordinate clause. Subordinate clauses usually start with a conjunction. You'll see that the given clause starts with "while", which is a conjunction.
This is an independent clause. You can identify independent clauses when you read it and it has a full idea already. It can stand without another clause to back it up.
Second Clause: "while Mother takes a nap"
This is a subordinate clause. Subordinate clauses usually start with a conjunction. You'll see that the given clause starts with "while", which is a conjunction.
You would use a comma. The semi colon acts when both what is in front and behind it can stand alone as a sentence. “With technology in the hands of so many people” does not make sense standing alone.
