To rewrite \(\sqrt{63}\) as the product of an integer and another irrational square root, we first need to factor \(63\) into its prime components.
The prime factorization of \(63\) is: \[ 63 = 9 \times 7 = 3^2 \times 7 \]
Now we can utilize the property of square roots: \[ \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} \]
Since \(\sqrt{9} = 3\), we can simplify further: \[ \sqrt{63} = 3 \times \sqrt{7} \]
Thus, \(\sqrt{63}\) rewritten as the product of an integer and another irrational square root, in simplest form, is: \[ \boxed{3\sqrt{7}} \]