2005 AMC 10B Problem 4

Work from the inside of the parentheses out.

1. First inner part. $5 \diamond 12 = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
2. Second inner part. $(-12) \diamond (-5) = \sqrt{(-12)^2 + (-5)^2}$. A negative number times itself is positive, so $(-12)^2 = 144$ and $(-5)^2 = 25$. This is again $\sqrt{169} = 13$.
3. Outer step. Now the problem is $13 \diamond 13$:
$$13 \diamond 13 = \sqrt{13^2 + 13^2} = \sqrt{2 \cdot 13^2} = 13\sqrt{2}.$$
(Pull the $13^2$ out of the square root as $13$, and the $2$ stays inside.)

The answer is $\boxed{\textbf{(D)}\ 13\sqrt{2}}$.

A made-up symbol like $\diamond$ is just a rule: put the two numbers into the formula. With nested parentheses, do the innermost ones first and use their results in the outer step. It also helps to notice that $a \diamond a$ is always $a\sqrt{2}$, the diagonal of a square with side $a$.

Work from the inside out. Since squaring removes signs, the two inner operations are the same computation:
$$5 \diamond 12 = \sqrt{25 + 144} = \sqrt{169} = 13, \qquad (-12) \diamond (-5) = \sqrt{144 + 25} = 13.$$

Now apply the operation once more:
$$13 \diamond 13 = \sqrt{13^2 + 13^2} = \sqrt{2 \cdot 169} = 13\sqrt{2}.$$

The answer is $\boxed{\textbf{(D)}\ 13\sqrt{2}}$.