To simplify the expression \((19^4)^9\) using the power rule of exponents, you apply the rule that states \((a^m)^n = a^{m \cdot n}\).
In this case, you have:
\[ (19^4)^9 = 19^{4 \cdot 9} \]
Now, calculate \(4 \cdot 9\):
\[ 4 \cdot 9 = 36 \]
Therefore, the simplified expression is:
\[ 19^{36} \]