To multiply \( (5\sqrt{19})(6\sqrt{2}) \), we can multiply the coefficients and the square roots separately:
\[ 5 \cdot 6 = 30 \]
Next, we multiply the square roots:
\[ \sqrt{19} \cdot \sqrt{2} = \sqrt{19 \cdot 2} = \sqrt{38} \]
Now, we can combine the two results:
\[ (5\sqrt{19})(6\sqrt{2}) = 30\sqrt{38} \]
Thus, the exact answer is:
\[ \boxed{30\sqrt{38}} \]