If 7 * 24 = 25 and 12 * 16 = 20, then what is 16 * 63 equal to ?

The correct answer is 65.

To solve this problem, we need to identify the pattern or logic behind the custom operator "*".

Let's analyze the given equations:

1. 7 * 24 = 25
Notice that 7, 24, and 25 form a Pythagorean triple, where:
$7^2+{24}^2=49+576=625$
Taking the square root of 625:
$\sqrt{625}=25$

2. 12 * 16 = 20
Similarly, 12, 16, and 20 also follow the Pythagorean theorem:
${12}^2+{16}^2=144+256=400$
Taking the square root of 400:
$\sqrt{400}=20$

Thus, the operation a * b is defined as:
$a*b=\sqrt{a^2+b^2}$

Now, applying this rule to find the value of 16 * 63:
$16*63=\sqrt{{16}^2+{63}^2}$

Calculate the squares:
${16}^2=256$
${63}^2=3969$

Add the squares together:
$256+3969=4225$

Finally, find the square root:
$\sqrt{4225}=65$

Therefore, 16 * 63 is equal to 65.