jmc

Using the given recurrence relation, we have $x_2=\frac{2}{x_1},$ so $x_1{x}_2=2.$ Similarly, $x_4=\frac{4}{x_3},$ so $x_4{x}_3=4,$ and $x_6{x}_5=6,$ $x_8{x}_7=8.$ Therefore, $$x_1{x}_2\cdots x_8=(x_1{x}_2)(x_3{x}_4)(x_5{x}_6)(x_7{x}_8)=2\cdot 4\cdot 6\cdot 8=\boxed{384}.$$(Note that the initial value $x_1=97$ was unnecessary.)