prime factors

The wording of the question is not good - where is the question from? When we talk about the prime factors of a number, normally we mean the number of distinct prime factors. That is, 30 has three prime factors: 2, 3 and 5, and 900 also has three prime factors - 2, 3 and 5. The powers in the prime factorization aren't relevant if you're only counting prime factors.

The question really means to ask for what is known in Number Theory as the 'length' of 30^7 * 22^5 * 34^11. The question should be asking "How many primes would need to be multiplied together to get 30^7 * 22^5 * 34^11"? To answer that, we simply need to prime factorize the number, and add together the powers.

I'd add that you don't often see questions that ask you to do this on the GMAT, though there is one in the Official Guide (that's the only one I've seen, in fact).