5^11 x 4^5 = 5 x 10^k

this can also be written as 5 x 5^10 x 4^5
or 5 x 5^10 x (2^2)^5
or 5 x 5^10 x 2^10
or 5 x (5 x 2)^10
or 5 x 10^10
thus k=10

realize that you can multiply distinct bases that have the same exponent together and get a new number that has the same exponent - for example - 2^2(3^2) = 6^2

Now to get to a 10 as a base you need a 5 and a 2 - you can work with the first part of the equation or the second - since Neo worked with the first, I'll show you how to work with the second.
5 x 10^k = 5 x 5^k x 2^k = 5^(k+1) x 4^k = 5^(k+1) x 4^(k/2)

On the other side we have 5^11 x 4^5 so we know k+1= 11 and k/2=5 thus for each k = 10.

The skill involved here is factoring and multiplying bases