To find the product of \( 5\sqrt{18} \) and \( 5\sqrt{18} \), we can express it as:
\[ (5\sqrt{18}) \times (5\sqrt{18}) \]
This simplifies to:
\[ 5 \times 5 \times \sqrt{18} \times \sqrt{18} = 25 \times (\sqrt{18})^2 \]
Since \((\sqrt{18})^2\) is simply \(18\), we can rewrite the expression as:
\[ 25 \times 18 \]
Now, calculating this gives:
\[ 25 \times 18 = 450 \]
Next, we can conclude whether \(450\) is rational or irrational. A number is considered rational if it can be expressed as the quotient of two integers. Since \(450\) is an integer, it is clearly rational.
Thus, the final product is:
\[ \boxed{450} \]
and the result is rational since \(450\) is an integer.