To multiply \(5\sqrt{19}\) and \(6\sqrt{2}\), you can use the properties of multiplication:
\[ (5\sqrt{19})(6\sqrt{2}) = (5 \times 6)(\sqrt{19} \times \sqrt{2}) \]
Calculating the coefficients:
\[ 5 \times 6 = 30 \]
Now, multiplying the square roots:
\[ \sqrt{19} \times \sqrt{2} = \sqrt{19 \times 2} = \sqrt{38} \]
Putting it all together:
\[ (5\sqrt{19})(6\sqrt{2}) = 30\sqrt{38} \]
So, the exact number is:
\[ 30\sqrt{38} \]