How do you know when to simplify further when simplifying radicals?

Question

I’m not sure how to phrase this, but say you are simplifying √128 and solve it as such:

√128 = √16 √8 = 4√16

Now, you can see that the process is correct, but the student did not use the highest perfect square that can go into √128. My question is, what does the student need to do to identify that the radical can be simplified further, simplify the radical so that it is the correct answer, and/or find a strategy so that the student can find the biggest perfect square in every radical?

Answer 1

Answer 2

Well the process you present is not correct√

√128 = √16 √8

√16 √8 = 4 √8

One process is to attempt dividing any remaining radicand with a perfect square, starting with the smalled, 4

8 is divisible by 4, so √8 = √4 √2.

Does that answer?