8.4. It is known that $a^{2}+2=b^{4}, b^{2}+2=c^{4}, c^{2}+2=a^{4}$. What is the value of the product $\left(a^{2}-1\right)\left(b^{2}-1\right)\left(c^{2}-1\right)$? Find all possible values and prove that there are no others.
Solution: Subtract 1 from both sides of each equation, we get:

$$
\begin{gathered}
a^{2}+1=b^{4}-1=\left(b^{2}-1\right)\left(b^{2}+1\right) \\
b^{2}+1=\left(c^{2}-1\right)\left(c^{2}+1\right), \quad c^{2}+1=\left(a^{2}-1\right)\left(a^{2}+1\right)
\end{gathered}
$$

Multiply the left and right parts of all three equations, and after canceling the positive number $\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)$, we get $\left(a^{2}-1\right)\left(b^{2}-1\right)\left(c^{2}-1\right)=1$. Therefore, the desired product can only be 1. Since the transformations involved an inequivalent step (multiplying the left and right parts of several equations is only a transition to a consequence), it is necessary to show that the number 1 can indeed be obtained. This case will occur, for example, when $a=b=c=\sqrt{2}$.

Answer: 1.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer with a confirming example of numbers $a, b, c$ | 7 points |
| Correct and justified answer without a confirming example of numbers $a, b, c$ | 6 points |
| Example of numbers $a, b, c$ satisfying the condition of the problem, for which the value of the desired expression is 1 | 1 point |
| Algebraic calculations that did not lead to a solution, as well as an answer without justification or an incorrect answer | 0 points |