Question

对实数 $a$ 和 $b$ ，定义 $a \diamond b = \sqrt{a^2 + b^2}$ 。求的值。 $(5 \diamond 12) \diamond ((-12) \diamond (-5))?$

For real numbers $a$ and $b,$ define $a \diamond b = \sqrt{a^2 + b^2}.$ What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))?$

$0$

$\dfrac{17}{2}$

$13$

$13\sqrt{2}$

$26$

两个内层表达式分别为且同理 $(-12) \diamond (-5)$ $= \sqrt{144 + 25} = 13$ 。 $5 \diamond 12 = \sqrt{5^2 + 12^2} = 13$

因此 $13 \diamond 13 = \sqrt{13^2 + 13^2} = 13\sqrt{2}.$

所以正确答案是 D。

Each inner expression evaluates to $5 \diamond 12 = \sqrt{5^2 + 12^2} = 13$ and similarly $(-12) \diamond (-5)$ $= \sqrt{144 + 25} = 13.$

Then $13 \diamond 13 = \sqrt{13^2 + 13^2} = 13\sqrt{2}.$

Thus, D is the correct answer.