It depends on what you mean by "very strong", as not everyone's perspective is going to be the same.
I prefer to wait and split a hive when I have 3 deep boxes filled with bees and at least 10 deep frames of brood in various stages of development.
The number of splits I make depends on my goals. If I want that hive to produce honey, I may just take one split with the queen, leaving most of the brood to produce a honey crop 30-60 days from the split. If I am purchasing or raising queens for splits
that I'll grow up to over-winter or sell, I'll take 2 frames of emerging capped brood and extra nurse bees for each 5-frame nuc.
If I take 4 frames from a colony that has 12 deep frames of brood, it will reduce their urge to swarm for 2-3 weeks. Over that time, they will easily get back up to 12 deep frames of brood again, giving me the opportunity to either take the queen to a new
split or set up another two 5-frame nucleus colonies with queen cells or purchased queens. It's pretty easy to get two splits in May and another one or two splits in June from a strong hive, managing them to prevent swarming and still get honey production.
One strong hive can turn into 4 or 5 by July 1. If I don't care about honey production at all, I can probably get 2-3 more 5-frame nucs out of that colony over the same timeframe, which would be a total of 6-8 colonies by July 1.
I think the main mistake new beekeepers tend to make is splitting too early, before the colony has really reached the point where they have excess nurse bees, particularly when letting the bees raise their own queens. The other challenge in all of this is
managing drawn comb - having lots of extra drawn comb on hand is a huge benefit for being able to take frames and give the queen new space to lay. The need to draw new comb will slow colony growth. Plus, you need to watch out for strong nectar flows - of
the colony doesn't have excess space to store nectar, they will back-fill the brood combs and accelerate the urge to swarm.
A lot of people over-winter in 2 deep boxes. If I were planning to split a hive I'd be focused right now (April) on getting a third box of deep frames drawn out. Bees are motivated to draw out new comb in spring and early summer, which will give you some
of the drawn comb that you need later.
Regards,
Joe