Little O notation relationship
One possibility is to merely apply the definition. That is, we see that if $\lim_{n \to \infty} f(n) / g(n) = 0$, then $f(n) = o(g(n))$. Computing this, we have that $$\lim_{n \to \infty} f(n) / g(n) = \lim_{n \to \infty} n^n/10^{10n} = \infty \neq 0.$$ We conclude that $f(n) = o(g(n))$ does not hold.