Interpret the operation “*” as mapping two legs of a right triangle to its hypotenuse using the Pythagorean theorem:

For a right triangle with legs a and b, $a^2 + b^2 = c^2$ ⇒ $c = \sqrt{a^2 + b^2}$

Given checks:

• 7 * 24 = $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$
• 12 * 16 = $\sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20$

Required: 16 * 63 = $\sqrt{16^2 + 63^2} = \sqrt{256 + 3969} = \sqrt{4225} = 65$

Therefore, 16 * 63 = $\textbf{65}$.