Solution:
5^21 * 4^11 = 5^21 * 2^22 = 5^21 * 2^21 * 2 = 10^21 * 2.
So n = 21.

Wow, thanks, that is so much simpler than the way I solved it. What is the best way to quickly see a solution like that? It took me a while to figure it out. Definitely not in 2 mins.
How would you solve this problem?

The question has 2*(10^n). So obviously you will need pairs of 2's and 5's because they will combine to give 10.
Try to break up 5^21 * 4^11 so that you get equal powers of 2's and 5's with other factors.
Now 4^11 is 2^22. This has 22 2's.
Now 5^21 has 21 5's.
So when we pair up, we get 21 pairs of 2's and 5's and one 2 extra.
5^21 * 4^11 = 5^21 * 2^22 = 5^21 * 2^21 * 2 = 10^21 * 2.
So n = 21.