For each pair of real numbers $a \ne b$, define the operation $\star$ as \[
(a \star b) = \frac{a + b}{a - b}.
\]What is the value of $((1 \star 2) \star 3)$?
First we have \[
(1 \star 2) = \frac{1 + 2}{1 - 2} = -3.
\]Then \[
((1 \star 2) \star 3) = (-3 \star 3) = \frac{-3 + 3}{-3 - 3} = \frac{0}{-6} = \boxed{0}.
\]
0
Level 2
Algebra
Problem: For each pair of real numbers $a \ne b$, define the operation $\star$ as \[
(a \star b) = \frac{a + b}{a - b}.
\]What is the value of $((1 \star 2) \star 3)$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the value of $((1 \star 2) \star 3)$, we first need to compute $1 \star 2$. Using the definition of the operation $\star$, we have:

1. Calculate $1 \star 2$:
   \[
   1 \star 2 = \frac{1 + 2}{1 - 2} = \frac{3}{-1} = -3.
   \]

2. Next, we need to compute $(-3) \star 3$:
   \[
   (-3) \star 3 = \frac{-3 + 3}{-3 - 3} = \frac{0}{-6} = 0.
   \]

Thus, the value of $((1 \star 2) \star 3)$ is $0$.<|END_OF_LATENT|> The final answer is $0$.