To rewrite \(\sqrt{63}\) as the product of an integer and another irrational square root, we can start by factoring 63 into its prime factors:
\[ 63 = 9 \times 7 = 3^2 \times 7 \]
Now, we can take the square root:
\[ \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} \]
Since \(\sqrt{9} = 3\), we can simplify this further:
\[ \sqrt{63} = 3 \times \sqrt{7} \]
Thus, the simplest form of \(\sqrt{63}\) rewritten as the product of an integer and another irrational square root is:
\[ 3\sqrt{7} \]