This is nice because 4 is a perfect square and the only one hiding inside our radicand. It wasn't doing a very good job, obviously, since we found it right away. Given this,
We can't simplify any more. 30 factors completely as
2 × 3 × 5
and no factor of 30 is a perfect square, so we're done. The nicest it ever looks is
This one's a beast. It's such a big number that it even has a comma. We want to make this bugger more manageable, so we'll factor it first. Remember, we're looking for factors that are perfect squares. There's a glaring one here from that list you memorized. Can you spy it with your little eye?
Rearrange the factors a bit and write
39,600 = 100 × (2 × 3) × (2 × 3) × 11.
Now we can break up the radical by "unmultiplying." With any luck, it will result in a term that gives us an "unheadache."
Finally, we replace the radicals of perfect squares with their square roots:
A radical term is considered simplified when there is only one radical sign and the radicand has no factors that are perfect squares. This is the case above, so we can call this mission accomplished. We don't even remember choosing to accept it.
We can factor 12 as 4 × 3:
Then, we can multiply the remaining radicals and factor once more:
We can break this up into .
We separate them right away in case a brawl is about to break out. Better step in before things get bloody.