To simplify the expression

\[
\frac{4^{10^{10}} \cdot 9^{10^{10}}}{3^{2019} \cdot 16^{504}},
\]

let's rewrite each term in the expression using base powers:

1. Rewrite \(4\), \(9\), and \(16\) in terms of their prime factors:
- \(4 = 2^2\)
- \(9 = 3^2\)
- \(16 = 2^4\)

Now substituting these into the expression, we have:

\[
4^{10^{10}} = (2^2)^{10^{10}} = 2^{2 \cdot 10^{10}} = 2^{2 \times 10^{10}},
\]

\[
9^{10^{10}} = (3^2)^{10^{10}} = 3^{2 \cdot 10^{10}} = 3^{2 \times 10^{10}},
\]

\[
16^{504} = (2^4)^{504} = 2^{4 \cdot 504} = 2^{2016}.
\]

Now substituting these into the original expression:

\[
\frac{2^{2 \times 10^{10}} \cdot 3^{2 \times 10^{10}}}{3^{2019} \cdot 2^{2016}}.
\]

Next, we can separate the powers of \(2\) and \(3\):

\[
= \frac{2^{2 \times 10^{10}}}{2^{2016}} \cdot \frac{3^{2 \times 10^{10}}}{3^{2019}}.
\]

Utilizing the properties of exponents, we simplify:

\[
= 2^{2 \times 10^{10} - 2016} \cdot 3^{2 \times 10^{10} - 2019}.
\]

Now, we can express the final result as:

\[
= 2^{2 \times 10^{10} - 2016} \cdot 3^{2 \times 10^{10} - 2019}.
\]

This expression simplifies the original fraction in terms of its prime factors.