Для упрощения выражения сначала выполним операции с дробями.

12a + 9b / x^2 - 81 : 8a + 6b / x^2 + 18x + 81 =
= 12a + 9b / (x - 9)(x + 9) : 8a + 6b / (x + 9)(x + 9) =
= 12a + 9b / (x - 9)(x + 9) : 8a + 6b / (x + 9)^2 =
= (12a(x + 9) + 9b) / (x - 9)(x + 9) : (8a(x + 9) + 6b) / (x + 9)^2 =
= ((12a(x + 9) + 9b) (x + 9)^2) / ((x - 9)(x + 9) (8a(x + 9) + 6b)) =
= ((12a(1/2 + 9) + 9(-1/6)) (1/2 + 9)^2) / ((1/2 - 9)(1/2 + 9) (8a(1/2 + 9) + 6b)) =
= ((12(1/7)(1/2 + 9) - 9/6) (1/2 + 9)^2) / ((1/2 - 9)(1/2 + 9) (8(1/7)(1/2 + 9) + 6(-1/6))) =
= ((12/7 19/2 - 3/2) 19/2) / ((-17/2) 19/2 (8/7 19/2 - 1)) =
= ((228/14 19/2 - 3/2) 19/2) / (-17/2 19/2 152/14 - 1)) =
= ((26 - 3/2) 19/2) / (289/2 152/14 - 1) =
= (49/2 19/2) / (1444/14 - 1) =
= 2353/56 / (1444/14 - 1) =
= 2353/56 / (1030/7) =
= 2353/56 7/1030 =
= 11829/74240

Подставляя a = 1/7, b = -1/6, x = 1/2 в полученное упрощенное выражение, получим:

11829/74240 ≈ 0.1593

Ответ: 0.1593.