Please show your calcs...
That one is easy.
Fill in the first equation with the content of the second. You get:
(10^100)^(10^100) = 10^k
The crux is the following. On both sides of the equation you have 10 as the base, you must now find out k. Simplify:
(10^10^2) ^ (10^100) = 10^k
Now you can multiply the exponents.
10^(10^2 * 10^100) = 10 ^k
Focus on the equation inside the parentheses. Since you have the same base you can add 2 and 100. The result is 10^102